Mixing
Double sum over $leftlfloor{ac+bdover k}rightrfloor$
$begingroup$
We have
$$leftlfloor{ac+bdover k}rightrfloor-leftlfloor{ac+bd-1over k}rightrfloor=1-leftlceil{ (ac+bd)mod{k}over k}rightrceil$$
for $a,b,c,d,k$ - integers, $ageqslant0$, $bgeqslant0$, $c>0$, $d>0$, $k>0$. Next we want to calculate
$$sumlimits_{a=0}^{n}sumlimits_{b=0}^{m}leftlfloor{ac+bdover k}rightrfloor$$
or
$$sumlimits_{a=0}^{n}sumlimits_{b=0}^{m}leftlceil{(ac+bd)mod{k}over k}rightrceil$$
Is there a nice closed form for it?
summation closed-form floor-function ceiling-function
edited Jan 3 at 17:00
Key Flex
7,77461232
asked Jan 3 at 16:05
user514787user514787
715210
$endgroup$
add a comment |
$begingroup$
We have
$$leftlfloor{ac+bdover k}rightrfloor-leftlfloor{ac+bd-1over k}rightrfloor=1-leftlceil{ (ac+bd)mod{k}over k}rightrceil$$
for $a,b,c,d,k$ - integers, $ageqslant0$, $bgeqslant0$, $c>0$, $d>0$, $k>0$. Next we want to calculate
$$sumlimits_{a=0}^{n}sumlimits_{b=0}^{m}leftlfloor{ac+bdover k}rightrfloor$$
or
$$sumlimits_{a=0}^{n}sumlimits_{b=0}^{m}leftlceil{(ac+bd)mod{k}over k}rightrceil$$
Is there a nice closed form for it?
summation closed-form floor-function ceiling-function
edited Jan 3 at 17:00
Key Flex
7,77461232
asked Jan 3 at 16:05
user514787user514787
715210
$endgroup$
add a comment |
$begingroup$
We have
$$leftlfloor{ac+bdover k}rightrfloor-leftlfloor{ac+bd-1over k}rightrfloor=1-leftlceil{ (ac+bd)mod{k}over k}rightrceil$$
for $a,b,c,d,k$ - integers, $ageqslant0$, $bgeqslant0$, $c>0$, $d>0$, $k>0$. Next we want to calculate
$$sumlimits_{a=0}^{n}sumlimits_{b=0}^{m}leftlfloor{ac+bdover k}rightrfloor$$
or
$$sumlimits_{a=0}^{n}sumlimits_{b=0}^{m}leftlceil{(ac+bd)mod{k}over k}rightrceil$$
Is there a nice closed form for it?
summation closed-form floor-function ceiling-function
edited Jan 3 at 17:00
Key Flex
7,77461232
asked Jan 3 at 16:05
user514787user514787
715210
$endgroup$
We have
$$leftlfloor{ac+bdover k}rightrfloor-leftlfloor{ac+bd-1over k}rightrfloor=1-leftlceil{ (ac+bd)mod{k}over k}rightrceil$$
for $a,b,c,d,k$ - integers, $ageqslant0$, $bgeqslant0$, $c>0$, $d>0$, $k>0$. Next we want to calculate
$$sumlimits_{a=0}^{n}sumlimits_{b=0}^{m}leftlfloor{ac+bdover k}rightrfloor$$
or
$$sumlimits_{a=0}^{n}sumlimits_{b=0}^{m}leftlceil{(ac+bd)mod{k}over k}rightrceil$$
Is there a nice closed form for it?
summation closed-form floor-function ceiling-function
summation closed-form floor-function ceiling-function
edited Jan 3 at 17:00
Key Flex
7,77461232
asked Jan 3 at 16:05
user514787user514787
715210
edited Jan 3 at 17:00
Key Flex
7,77461232
asked Jan 3 at 16:05
user514787user514787
715210
edited Jan 3 at 17:00
Key Flex
7,77461232
edited Jan 3 at 17:00
Key Flex
7,77461232
edited Jan 3 at 17:00
Key Flex
7,77461232
7,77461232
asked Jan 3 at 16:05
user514787user514787
715210
asked Jan 3 at 16:05
user514787user514787
715210
asked Jan 3 at 16:05
user514787user514787
715210
715210
add a comment |
add a comment |
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