Mixing
Have I discovered a new significance to a previously discovered constant?
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I've been interested in infinite sums for a while, though I have no formal education of them. I was messing around with repeated division and addition (e.g. 1 + (1 / (1 + (1 /...)))) I then plugged fibonacci numbers into the above pattern, and as I calculated more and more layers, the result converged to around 1.39418655, which I just now found out was a constant called Madachy's constant. However, from the research that I did on the constant (I found very little), it doesn't seem to be related to infinite series or Fibonacci numbers at all. Have I found a new way to calculate this number? Does it give it any more significance? 
fibonacci-numbers constants infinitesimals
asked Jan 18 at 4:31
Matthew AverillMatthew Averill
253
$endgroup$
add a comment |
$begingroup$
I've been interested in infinite sums for a while, though I have no formal education of them. I was messing around with repeated division and addition (e.g. 1 + (1 / (1 + (1 /...)))) I then plugged fibonacci numbers into the above pattern, and as I calculated more and more layers, the result converged to around 1.39418655, which I just now found out was a constant called Madachy's constant. However, from the research that I did on the constant (I found very little), it doesn't seem to be related to infinite series or Fibonacci numbers at all. Have I found a new way to calculate this number? Does it give it any more significance?
fibonacci-numbers constants infinitesimals
asked Jan 18 at 4:31
Matthew AverillMatthew Averill
253
$endgroup$
3
$begingroup$
Did you see this? fq.math.ca/Scanned/6-6/madachy.pdf
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– Cheerful Parsnip
Jan 18 at 4:58
1
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Wow, that sounds like an answer to me ...
$endgroup$
– kcrisman
Jan 18 at 5:00
$begingroup$
Also found in the OEIS as A130701 which also has a link to the above paper.
$endgroup$
– Tito Piezas III
Jan 19 at 3:28
add a comment |
$begingroup$
I've been interested in infinite sums for a while, though I have no formal education of them. I was messing around with repeated division and addition (e.g. 1 + (1 / (1 + (1 /...)))) I then plugged fibonacci numbers into the above pattern, and as I calculated more and more layers, the result converged to around 1.39418655, which I just now found out was a constant called Madachy's constant. However, from the research that I did on the constant (I found very little), it doesn't seem to be related to infinite series or Fibonacci numbers at all. Have I found a new way to calculate this number? Does it give it any more significance?
fibonacci-numbers constants infinitesimals
asked Jan 18 at 4:31
Matthew AverillMatthew Averill
253
$endgroup$
I've been interested in infinite sums for a while, though I have no formal education of them. I was messing around with repeated division and addition (e.g. 1 + (1 / (1 + (1 /...)))) I then plugged fibonacci numbers into the above pattern, and as I calculated more and more layers, the result converged to around 1.39418655, which I just now found out was a constant called Madachy's constant. However, from the research that I did on the constant (I found very little), it doesn't seem to be related to infinite series or Fibonacci numbers at all. Have I found a new way to calculate this number? Does it give it any more significance?
fibonacci-numbers constants infinitesimals
fibonacci-numbers constants infinitesimals
asked Jan 18 at 4:31
Matthew AverillMatthew Averill
253
asked Jan 18 at 4:31
Matthew AverillMatthew Averill
253
edited Jan 18 at 4:56
Matthew Averill
asked Jan 18 at 4:31
Matthew AverillMatthew Averill
253
asked Jan 18 at 4:31
Matthew AverillMatthew Averill
253
asked Jan 18 at 4:31
Matthew AverillMatthew Averill
253
253
3
$begingroup$
Did you see this? fq.math.ca/Scanned/6-6/madachy.pdf
$endgroup$
– Cheerful Parsnip
Jan 18 at 4:58
1
$begingroup$
Wow, that sounds like an answer to me ...
$endgroup$
– kcrisman
Jan 18 at 5:00
$begingroup$
Also found in the OEIS as A130701 which also has a link to the above paper.
$endgroup$
– Tito Piezas III
Jan 19 at 3:28
add a comment |
3
$begingroup$
Did you see this? fq.math.ca/Scanned/6-6/madachy.pdf
$endgroup$
– Cheerful Parsnip
Jan 18 at 4:58
1
$begingroup$
Wow, that sounds like an answer to me ...
$endgroup$
– kcrisman
Jan 18 at 5:00
$begingroup$
Also found in the OEIS as A130701 which also has a link to the above paper.
$endgroup$
– Tito Piezas III
Jan 19 at 3:28
3
3
$begingroup$
Did you see this? fq.math.ca/Scanned/6-6/madachy.pdf
$endgroup$
– Cheerful Parsnip
Jan 18 at 4:58
$begingroup$
Did you see this? fq.math.ca/Scanned/6-6/madachy.pdf
$endgroup$
– Cheerful Parsnip
Jan 18 at 4:58
1
1
$begingroup$
Wow, that sounds like an answer to me ...
$endgroup$
– kcrisman
Jan 18 at 5:00
$begingroup$
Wow, that sounds like an answer to me ...
$endgroup$
– kcrisman
Jan 18 at 5:00
$begingroup$
Also found in the OEIS as A130701 which also has a link to the above paper.
$endgroup$
– Tito Piezas III
Jan 19 at 3:28
$begingroup$
Also found in the OEIS as A130701 which also has a link to the above paper.
$endgroup$
– Tito Piezas III
Jan 19 at 3:28
add a comment |
3 Answers
3
active
oldest
votes
$begingroup$
Did you see this? fq.math.ca/Scanned/6-6/madachy.pdf It looks like it is exactly your calculation.
Friendly edit: This is in Fibonacci Quarterly, Volume 6 No. 6 page 385, as seen at the cover of that issue.
edited Jan 19 at 2:31
kcrisman
1,5401020
answered Jan 18 at 5:01
Cheerful ParsnipCheerful Parsnip
21.1k23598
$endgroup$
$begingroup$
It seems I did not, I could only find stuff about magic squares. Thank you!
$endgroup$
– Matthew Averill
Jan 18 at 5:02
$begingroup$
By the way @Cheerful_Parsnip - can you add a more proper reference for this? Presumably it appeared in some publication, and that would add a lot of value to this answer - surely it will come up again!
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– kcrisman
Jan 18 at 5:03
1
$begingroup$
@Kcrisman, if it were ever published, it's not indexed by MathSciNet. I found it by googling "Madachy constant."
$endgroup$
– Cheerful Parsnip
Jan 18 at 18:56
$begingroup$
"Readers of this Journal will note immediately that the terms of this continued fraction are successive Fibonacci terms." That implies publication - got it: Unsurprisingly, Fibonacci Quarterly (which might not be indexed by MR, true) scribd.com/document/270530286/6-6 - oh, duh! the domain name IS the Fibonacci Quarterly - fq.math.ca! Actually it is indexed by MSN mathscinet.ams.org/mathscinet/search/…
$endgroup$
– kcrisman
Jan 19 at 2:11
1
$begingroup$
Hope the friendly edit is okay
$endgroup$
– kcrisman
Jan 19 at 2:14
|
show 1 more comment
$begingroup$
(Too long for a comment.) The OP's continued fraction which uses the Fibonacci numbers $F_n$ is
$$mu =1+cfrac{1}{2 + cfrac{3} {5 + cfrac{8} {13 + cfrac{21} {34+ddots}}}}=1.3941865dots $$
which, as pointed out by Cheerful Parsnip, was previously investigated by Joseph Madachy and whose constant is designed by OEIS A130701 as $mu$.
However, there is continued fraction whose terms are also Fibonacci numbers and has a known closed-form,
$$frac{F_{n+2}}{F_{n+1}} =1^2+cfrac{(1cdot1)^2} {1^2+ cfrac{(1cdot2)^2} {2^2 + cfrac{(2cdot3)^2} {3^2 + cfrac{(3cdot5)^2} {5^2 + cfrac{(5cdot8)^2} {F_n^2 + cfrac{(F_{n}cdot F_{n+1})^2} {F_{n+1}^2+ddots} }}}}}$$
If we truncate it at the $n$th term, we get that ratio. Note $lim frac{F_{n+1}}{F_{n}} = phi$ as $ntoinfty$.
P.S. It would be nice if $mu$ has a closed-form in terms of the golden ratio $phi$ as well.
answered Jan 19 at 5:39
Tito Piezas IIITito Piezas III
27.3k366174
$endgroup$
$begingroup$
It won't let me make a one-character edit, but note the trivial potential edit that your spell checker put an "l" in Madachy's name.
$endgroup$
– kcrisman
Jan 19 at 14:02
1
$begingroup$
@kcrisman: Thanks. Has been fixed.
$endgroup$
– Tito Piezas III
Jan 19 at 14:50
add a comment |
$begingroup$
The continued fraction you are calculating will converge to the golden ratio. It is most certainly a significant constant having history spanning several millennia.
answered Jan 18 at 4:42
CyclotomicFieldCyclotomicField
2,4431314
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3
$begingroup$
The "all $1$s" continued fraction does indeed converge to the Golden Ratio. However, OP is interested in the continued fraction whose terms are the Fibonacci numbers.
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– Blue
Jan 18 at 4:51
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But the golden ratio converges on the first continued fraction I mentioned, with only ones. With Fibonacci numbers, the fraction converges to a number ~.3 less than the golden ratio. The work I linked shows converging that starts with boundaries 1 and 1.5, so it couldn't end up converging to 1.618. Edit: just saw the reply above as I was typing this
$endgroup$
– Matthew Averill
Jan 18 at 4:53
add a comment |
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3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Did you see this? fq.math.ca/Scanned/6-6/madachy.pdf It looks like it is exactly your calculation.
Friendly edit: This is in Fibonacci Quarterly, Volume 6 No. 6 page 385, as seen at the cover of that issue.
edited Jan 19 at 2:31
kcrisman
1,5401020
answered Jan 18 at 5:01
Cheerful ParsnipCheerful Parsnip
21.1k23598
$endgroup$
$begingroup$
It seems I did not, I could only find stuff about magic squares. Thank you!
$endgroup$
– Matthew Averill
Jan 18 at 5:02
$begingroup$
By the way @Cheerful_Parsnip - can you add a more proper reference for this? Presumably it appeared in some publication, and that would add a lot of value to this answer - surely it will come up again!
$endgroup$
– kcrisman
Jan 18 at 5:03
1
$begingroup$
@Kcrisman, if it were ever published, it's not indexed by MathSciNet. I found it by googling "Madachy constant."
$endgroup$
– Cheerful Parsnip
Jan 18 at 18:56
$begingroup$
"Readers of this Journal will note immediately that the terms of this continued fraction are successive Fibonacci terms." That implies publication - got it: Unsurprisingly, Fibonacci Quarterly (which might not be indexed by MR, true) scribd.com/document/270530286/6-6 - oh, duh! the domain name IS the Fibonacci Quarterly - fq.math.ca! Actually it is indexed by MSN mathscinet.ams.org/mathscinet/search/…
$endgroup$
– kcrisman
Jan 19 at 2:11
1
$begingroup$
Hope the friendly edit is okay
$endgroup$
– kcrisman
Jan 19 at 2:14
|
show 1 more comment
$begingroup$
Did you see this? fq.math.ca/Scanned/6-6/madachy.pdf It looks like it is exactly your calculation.
Friendly edit: This is in Fibonacci Quarterly, Volume 6 No. 6 page 385, as seen at the cover of that issue.
edited Jan 19 at 2:31
kcrisman
1,5401020
answered Jan 18 at 5:01
Cheerful ParsnipCheerful Parsnip
21.1k23598
$endgroup$
$begingroup$
It seems I did not, I could only find stuff about magic squares. Thank you!
$endgroup$
– Matthew Averill
Jan 18 at 5:02
$begingroup$
By the way @Cheerful_Parsnip - can you add a more proper reference for this? Presumably it appeared in some publication, and that would add a lot of value to this answer - surely it will come up again!
$endgroup$
– kcrisman
Jan 18 at 5:03
1
$begingroup$
@Kcrisman, if it were ever published, it's not indexed by MathSciNet. I found it by googling "Madachy constant."
$endgroup$
– Cheerful Parsnip
Jan 18 at 18:56
$begingroup$
"Readers of this Journal will note immediately that the terms of this continued fraction are successive Fibonacci terms." That implies publication - got it: Unsurprisingly, Fibonacci Quarterly (which might not be indexed by MR, true) scribd.com/document/270530286/6-6 - oh, duh! the domain name IS the Fibonacci Quarterly - fq.math.ca! Actually it is indexed by MSN mathscinet.ams.org/mathscinet/search/…
$endgroup$
– kcrisman
Jan 19 at 2:11
1
$begingroup$
Hope the friendly edit is okay
$endgroup$
– kcrisman
Jan 19 at 2:14
|
show 1 more comment
$begingroup$
Did you see this? fq.math.ca/Scanned/6-6/madachy.pdf It looks like it is exactly your calculation.
Friendly edit: This is in Fibonacci Quarterly, Volume 6 No. 6 page 385, as seen at the cover of that issue.
edited Jan 19 at 2:31
kcrisman
1,5401020
answered Jan 18 at 5:01
Cheerful ParsnipCheerful Parsnip
21.1k23598
$endgroup$
Did you see this? fq.math.ca/Scanned/6-6/madachy.pdf It looks like it is exactly your calculation.
Friendly edit: This is in Fibonacci Quarterly, Volume 6 No. 6 page 385, as seen at the cover of that issue.
edited Jan 19 at 2:31
kcrisman
1,5401020
answered Jan 18 at 5:01
Cheerful ParsnipCheerful Parsnip
21.1k23598
edited Jan 19 at 2:31
kcrisman
1,5401020
edited Jan 19 at 2:31
kcrisman
1,5401020
edited Jan 19 at 2:31
kcrisman
1,5401020
1,5401020
answered Jan 18 at 5:01
Cheerful ParsnipCheerful Parsnip
21.1k23598
answered Jan 18 at 5:01
Cheerful ParsnipCheerful Parsnip
21.1k23598
answered Jan 18 at 5:01
Cheerful ParsnipCheerful Parsnip
21.1k23598
21.1k23598
$begingroup$
It seems I did not, I could only find stuff about magic squares. Thank you!
$endgroup$
– Matthew Averill
Jan 18 at 5:02
$begingroup$
By the way @Cheerful_Parsnip - can you add a more proper reference for this? Presumably it appeared in some publication, and that would add a lot of value to this answer - surely it will come up again!
$endgroup$
– kcrisman
Jan 18 at 5:03
1
$begingroup$
@Kcrisman, if it were ever published, it's not indexed by MathSciNet. I found it by googling "Madachy constant."
$endgroup$
– Cheerful Parsnip
Jan 18 at 18:56
$begingroup$
"Readers of this Journal will note immediately that the terms of this continued fraction are successive Fibonacci terms." That implies publication - got it: Unsurprisingly, Fibonacci Quarterly (which might not be indexed by MR, true) scribd.com/document/270530286/6-6 - oh, duh! the domain name IS the Fibonacci Quarterly - fq.math.ca! Actually it is indexed by MSN mathscinet.ams.org/mathscinet/search/…
$endgroup$
– kcrisman
Jan 19 at 2:11
1
$begingroup$
Hope the friendly edit is okay
$endgroup$
– kcrisman
Jan 19 at 2:14
|
show 1 more comment
$begingroup$
It seems I did not, I could only find stuff about magic squares. Thank you!
$endgroup$
– Matthew Averill
Jan 18 at 5:02
$begingroup$
By the way @Cheerful_Parsnip - can you add a more proper reference for this? Presumably it appeared in some publication, and that would add a lot of value to this answer - surely it will come up again!
$endgroup$
– kcrisman
Jan 18 at 5:03
1
$begingroup$
@Kcrisman, if it were ever published, it's not indexed by MathSciNet. I found it by googling "Madachy constant."
$endgroup$
– Cheerful Parsnip
Jan 18 at 18:56
$begingroup$
"Readers of this Journal will note immediately that the terms of this continued fraction are successive Fibonacci terms." That implies publication - got it: Unsurprisingly, Fibonacci Quarterly (which might not be indexed by MR, true) scribd.com/document/270530286/6-6 - oh, duh! the domain name IS the Fibonacci Quarterly - fq.math.ca! Actually it is indexed by MSN mathscinet.ams.org/mathscinet/search/…
$endgroup$
– kcrisman
Jan 19 at 2:11
1
$begingroup$
Hope the friendly edit is okay
$endgroup$
– kcrisman
Jan 19 at 2:14
$begingroup$
It seems I did not, I could only find stuff about magic squares. Thank you!
$endgroup$
– Matthew Averill
Jan 18 at 5:02
$begingroup$
It seems I did not, I could only find stuff about magic squares. Thank you!
$endgroup$
– Matthew Averill
Jan 18 at 5:02
$begingroup$
By the way @Cheerful_Parsnip - can you add a more proper reference for this? Presumably it appeared in some publication, and that would add a lot of value to this answer - surely it will come up again!
$endgroup$
– kcrisman
Jan 18 at 5:03
$begingroup$
By the way @Cheerful_Parsnip - can you add a more proper reference for this? Presumably it appeared in some publication, and that would add a lot of value to this answer - surely it will come up again!
$endgroup$
– kcrisman
Jan 18 at 5:03
1
1
$begingroup$
@Kcrisman, if it were ever published, it's not indexed by MathSciNet. I found it by googling "Madachy constant."
$endgroup$
– Cheerful Parsnip
Jan 18 at 18:56
$begingroup$
@Kcrisman, if it were ever published, it's not indexed by MathSciNet. I found it by googling "Madachy constant."
$endgroup$
– Cheerful Parsnip
Jan 18 at 18:56
$begingroup$
"Readers of this Journal will note immediately that the terms of this continued fraction are successive Fibonacci terms." That implies publication - got it: Unsurprisingly, Fibonacci Quarterly (which might not be indexed by MR, true) scribd.com/document/270530286/6-6 - oh, duh! the domain name IS the Fibonacci Quarterly - fq.math.ca! Actually it is indexed by MSN mathscinet.ams.org/mathscinet/search/…
$endgroup$
– kcrisman
Jan 19 at 2:11
$begingroup$
"Readers of this Journal will note immediately that the terms of this continued fraction are successive Fibonacci terms." That implies publication - got it: Unsurprisingly, Fibonacci Quarterly (which might not be indexed by MR, true) scribd.com/document/270530286/6-6 - oh, duh! the domain name IS the Fibonacci Quarterly - fq.math.ca! Actually it is indexed by MSN mathscinet.ams.org/mathscinet/search/…
$endgroup$
– kcrisman
Jan 19 at 2:11
1
1
$begingroup$
Hope the friendly edit is okay
$endgroup$
– kcrisman
Jan 19 at 2:14
$begingroup$
Hope the friendly edit is okay
$endgroup$
– kcrisman
Jan 19 at 2:14
|
show 1 more comment
$begingroup$
(Too long for a comment.) The OP's continued fraction which uses the Fibonacci numbers $F_n$ is
$$mu =1+cfrac{1}{2 + cfrac{3} {5 + cfrac{8} {13 + cfrac{21} {34+ddots}}}}=1.3941865dots $$
which, as pointed out by Cheerful Parsnip, was previously investigated by Joseph Madachy and whose constant is designed by OEIS A130701 as $mu$.
However, there is continued fraction whose terms are also Fibonacci numbers and has a known closed-form,
$$frac{F_{n+2}}{F_{n+1}} =1^2+cfrac{(1cdot1)^2} {1^2+ cfrac{(1cdot2)^2} {2^2 + cfrac{(2cdot3)^2} {3^2 + cfrac{(3cdot5)^2} {5^2 + cfrac{(5cdot8)^2} {F_n^2 + cfrac{(F_{n}cdot F_{n+1})^2} {F_{n+1}^2+ddots} }}}}}$$
If we truncate it at the $n$th term, we get that ratio. Note $lim frac{F_{n+1}}{F_{n}} = phi$ as $ntoinfty$.
P.S. It would be nice if $mu$ has a closed-form in terms of the golden ratio $phi$ as well.
answered Jan 19 at 5:39
Tito Piezas IIITito Piezas III
27.3k366174
$endgroup$
$begingroup$
It won't let me make a one-character edit, but note the trivial potential edit that your spell checker put an "l" in Madachy's name.
$endgroup$
– kcrisman
Jan 19 at 14:02
1
$begingroup$
@kcrisman: Thanks. Has been fixed.
$endgroup$
– Tito Piezas III
Jan 19 at 14:50
add a comment |
$begingroup$
(Too long for a comment.) The OP's continued fraction which uses the Fibonacci numbers $F_n$ is
$$mu =1+cfrac{1}{2 + cfrac{3} {5 + cfrac{8} {13 + cfrac{21} {34+ddots}}}}=1.3941865dots $$
which, as pointed out by Cheerful Parsnip, was previously investigated by Joseph Madachy and whose constant is designed by OEIS A130701 as $mu$.
However, there is continued fraction whose terms are also Fibonacci numbers and has a known closed-form,
$$frac{F_{n+2}}{F_{n+1}} =1^2+cfrac{(1cdot1)^2} {1^2+ cfrac{(1cdot2)^2} {2^2 + cfrac{(2cdot3)^2} {3^2 + cfrac{(3cdot5)^2} {5^2 + cfrac{(5cdot8)^2} {F_n^2 + cfrac{(F_{n}cdot F_{n+1})^2} {F_{n+1}^2+ddots} }}}}}$$
If we truncate it at the $n$th term, we get that ratio. Note $lim frac{F_{n+1}}{F_{n}} = phi$ as $ntoinfty$.
P.S. It would be nice if $mu$ has a closed-form in terms of the golden ratio $phi$ as well.
answered Jan 19 at 5:39
Tito Piezas IIITito Piezas III
27.3k366174
$endgroup$
$begingroup$
It won't let me make a one-character edit, but note the trivial potential edit that your spell checker put an "l" in Madachy's name.
$endgroup$
– kcrisman
Jan 19 at 14:02
1
$begingroup$
@kcrisman: Thanks. Has been fixed.
$endgroup$
– Tito Piezas III
Jan 19 at 14:50
add a comment |
$begingroup$
(Too long for a comment.) The OP's continued fraction which uses the Fibonacci numbers $F_n$ is
$$mu =1+cfrac{1}{2 + cfrac{3} {5 + cfrac{8} {13 + cfrac{21} {34+ddots}}}}=1.3941865dots $$
which, as pointed out by Cheerful Parsnip, was previously investigated by Joseph Madachy and whose constant is designed by OEIS A130701 as $mu$.
However, there is continued fraction whose terms are also Fibonacci numbers and has a known closed-form,
$$frac{F_{n+2}}{F_{n+1}} =1^2+cfrac{(1cdot1)^2} {1^2+ cfrac{(1cdot2)^2} {2^2 + cfrac{(2cdot3)^2} {3^2 + cfrac{(3cdot5)^2} {5^2 + cfrac{(5cdot8)^2} {F_n^2 + cfrac{(F_{n}cdot F_{n+1})^2} {F_{n+1}^2+ddots} }}}}}$$
If we truncate it at the $n$th term, we get that ratio. Note $lim frac{F_{n+1}}{F_{n}} = phi$ as $ntoinfty$.
P.S. It would be nice if $mu$ has a closed-form in terms of the golden ratio $phi$ as well.
answered Jan 19 at 5:39
Tito Piezas IIITito Piezas III
27.3k366174
$endgroup$
(Too long for a comment.) The OP's continued fraction which uses the Fibonacci numbers $F_n$ is
$$mu =1+cfrac{1}{2 + cfrac{3} {5 + cfrac{8} {13 + cfrac{21} {34+ddots}}}}=1.3941865dots $$
which, as pointed out by Cheerful Parsnip, was previously investigated by Joseph Madachy and whose constant is designed by OEIS A130701 as $mu$.
However, there is continued fraction whose terms are also Fibonacci numbers and has a known closed-form,
$$frac{F_{n+2}}{F_{n+1}} =1^2+cfrac{(1cdot1)^2} {1^2+ cfrac{(1cdot2)^2} {2^2 + cfrac{(2cdot3)^2} {3^2 + cfrac{(3cdot5)^2} {5^2 + cfrac{(5cdot8)^2} {F_n^2 + cfrac{(F_{n}cdot F_{n+1})^2} {F_{n+1}^2+ddots} }}}}}$$
If we truncate it at the $n$th term, we get that ratio. Note $lim frac{F_{n+1}}{F_{n}} = phi$ as $ntoinfty$.
P.S. It would be nice if $mu$ has a closed-form in terms of the golden ratio $phi$ as well.
answered Jan 19 at 5:39
Tito Piezas IIITito Piezas III
27.3k366174
edited Jan 19 at 14:50
answered Jan 19 at 5:39
Tito Piezas IIITito Piezas III
27.3k366174
answered Jan 19 at 5:39
Tito Piezas IIITito Piezas III
27.3k366174
answered Jan 19 at 5:39
Tito Piezas IIITito Piezas III
27.3k366174
27.3k366174
$begingroup$
It won't let me make a one-character edit, but note the trivial potential edit that your spell checker put an "l" in Madachy's name.
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– kcrisman
Jan 19 at 14:02
1
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@kcrisman: Thanks. Has been fixed.
$endgroup$
– Tito Piezas III
Jan 19 at 14:50
add a comment |
$begingroup$
It won't let me make a one-character edit, but note the trivial potential edit that your spell checker put an "l" in Madachy's name.
$endgroup$
– kcrisman
Jan 19 at 14:02
1
$begingroup$
@kcrisman: Thanks. Has been fixed.
$endgroup$
– Tito Piezas III
Jan 19 at 14:50
$begingroup$
It won't let me make a one-character edit, but note the trivial potential edit that your spell checker put an "l" in Madachy's name.
$endgroup$
– kcrisman
Jan 19 at 14:02
$begingroup$
It won't let me make a one-character edit, but note the trivial potential edit that your spell checker put an "l" in Madachy's name.
$endgroup$
– kcrisman
Jan 19 at 14:02
1
1
$begingroup$
@kcrisman: Thanks. Has been fixed.
$endgroup$
– Tito Piezas III
Jan 19 at 14:50
$begingroup$
@kcrisman: Thanks. Has been fixed.
$endgroup$
– Tito Piezas III
Jan 19 at 14:50
add a comment |
$begingroup$
The continued fraction you are calculating will converge to the golden ratio. It is most certainly a significant constant having history spanning several millennia.
answered Jan 18 at 4:42
CyclotomicFieldCyclotomicField
2,4431314
$endgroup$
3
$begingroup$
The "all $1$s" continued fraction does indeed converge to the Golden Ratio. However, OP is interested in the continued fraction whose terms are the Fibonacci numbers.
$endgroup$
– Blue
Jan 18 at 4:51
$begingroup$
But the golden ratio converges on the first continued fraction I mentioned, with only ones. With Fibonacci numbers, the fraction converges to a number ~.3 less than the golden ratio. The work I linked shows converging that starts with boundaries 1 and 1.5, so it couldn't end up converging to 1.618. Edit: just saw the reply above as I was typing this
$endgroup$
– Matthew Averill
Jan 18 at 4:53
add a comment |
$begingroup$
The continued fraction you are calculating will converge to the golden ratio. It is most certainly a significant constant having history spanning several millennia.
answered Jan 18 at 4:42
CyclotomicFieldCyclotomicField
2,4431314
$endgroup$
3
$begingroup$
The "all $1$s" continued fraction does indeed converge to the Golden Ratio. However, OP is interested in the continued fraction whose terms are the Fibonacci numbers.
$endgroup$
– Blue
Jan 18 at 4:51
$begingroup$
But the golden ratio converges on the first continued fraction I mentioned, with only ones. With Fibonacci numbers, the fraction converges to a number ~.3 less than the golden ratio. The work I linked shows converging that starts with boundaries 1 and 1.5, so it couldn't end up converging to 1.618. Edit: just saw the reply above as I was typing this
$endgroup$
– Matthew Averill
Jan 18 at 4:53
add a comment |
$begingroup$
The continued fraction you are calculating will converge to the golden ratio. It is most certainly a significant constant having history spanning several millennia.
answered Jan 18 at 4:42
CyclotomicFieldCyclotomicField
2,4431314
$endgroup$
The continued fraction you are calculating will converge to the golden ratio. It is most certainly a significant constant having history spanning several millennia.
answered Jan 18 at 4:42
CyclotomicFieldCyclotomicField
2,4431314
answered Jan 18 at 4:42
CyclotomicFieldCyclotomicField
2,4431314
answered Jan 18 at 4:42
CyclotomicFieldCyclotomicField
2,4431314
answered Jan 18 at 4:42
CyclotomicFieldCyclotomicField
2,4431314
2,4431314
3
$begingroup$
The "all $1$s" continued fraction does indeed converge to the Golden Ratio. However, OP is interested in the continued fraction whose terms are the Fibonacci numbers.
$endgroup$
– Blue
Jan 18 at 4:51
$begingroup$
But the golden ratio converges on the first continued fraction I mentioned, with only ones. With Fibonacci numbers, the fraction converges to a number ~.3 less than the golden ratio. The work I linked shows converging that starts with boundaries 1 and 1.5, so it couldn't end up converging to 1.618. Edit: just saw the reply above as I was typing this
$endgroup$
– Matthew Averill
Jan 18 at 4:53
add a comment |
3
$begingroup$
The "all $1$s" continued fraction does indeed converge to the Golden Ratio. However, OP is interested in the continued fraction whose terms are the Fibonacci numbers.
$endgroup$
– Blue
Jan 18 at 4:51
$begingroup$
But the golden ratio converges on the first continued fraction I mentioned, with only ones. With Fibonacci numbers, the fraction converges to a number ~.3 less than the golden ratio. The work I linked shows converging that starts with boundaries 1 and 1.5, so it couldn't end up converging to 1.618. Edit: just saw the reply above as I was typing this
$endgroup$
– Matthew Averill
Jan 18 at 4:53
3
3
$begingroup$
The "all $1$s" continued fraction does indeed converge to the Golden Ratio. However, OP is interested in the continued fraction whose terms are the Fibonacci numbers.
$endgroup$
– Blue
Jan 18 at 4:51
$begingroup$
The "all $1$s" continued fraction does indeed converge to the Golden Ratio. However, OP is interested in the continued fraction whose terms are the Fibonacci numbers.
$endgroup$
– Blue
Jan 18 at 4:51
$begingroup$
But the golden ratio converges on the first continued fraction I mentioned, with only ones. With Fibonacci numbers, the fraction converges to a number ~.3 less than the golden ratio. The work I linked shows converging that starts with boundaries 1 and 1.5, so it couldn't end up converging to 1.618. Edit: just saw the reply above as I was typing this
$endgroup$
– Matthew Averill
Jan 18 at 4:53
$begingroup$
But the golden ratio converges on the first continued fraction I mentioned, with only ones. With Fibonacci numbers, the fraction converges to a number ~.3 less than the golden ratio. The work I linked shows converging that starts with boundaries 1 and 1.5, so it couldn't end up converging to 1.618. Edit: just saw the reply above as I was typing this
$endgroup$
– Matthew Averill
Jan 18 at 4:53
add a comment |
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Did you see this? fq.math.ca/Scanned/6-6/madachy.pdf
$endgroup$
– Cheerful Parsnip
Jan 18 at 4:58
1
$begingroup$
Wow, that sounds like an answer to me ...
$endgroup$
– kcrisman
Jan 18 at 5:00
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Also found in the OEIS as A130701 which also has a link to the above paper.
$endgroup$
– Tito Piezas III
Jan 19 at 3:28