# [seqfan] Re: Number of solutions of a*b + c*d + ... + y*z = 0 (mod n)

Ron Hardin rhhardin at att.net
Mon Sep 6 18:14:15 CEST 2010

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----- Original Message ----
> From: Ron Hardin <rhhardin at att.net>
> To: seqfan at list.seqfan.eu
> Sent: Sun, September 5, 2010 9:10:26 PM
> Subject: [seqfan] Number of solutions of a*b + c*d + ... + y*z = 0 (mod n)
>
> How many distinct no-zero solutions are there of a sum of k products of  pairs
> of numbers in 1..n-1 such that the sum is zero (mod n)?
>
> These  don't seem to be in OEIS (computation is in progress so series will
>grow)
>
> distinct means solutions are commuted to lexicographical  order,
> b*a->a*b and  a*c+a*b->a*b+a*c
>
> all offset  1

Doing the same thing for sum = 1 (mod n) instead of zero (mod n) gives the table

..0....0......0.......0.........0..........0...........0............0
..1....0......1.......0.........1..........0...........1............0
..2....1......3.......7.........5..........9..........15...........12
..2....4.....14......30........64........112.........198..........318
..3....9.....46.....142.......400.......1004........2282.........4868
..2...19....106.....502......1914.......6404.......19300........53133
..4...31....254....1519......7589......32890......126861.......444019
..4...42....494....3828.....24902.....137528......670058......2935676
..4...75....939....9145.....73134.....499641.....2997685.....16112283
..3...91...1528...18966....188315....1579113....11539243.....75118525
..6..136...2668...38562....455127....4551268....39660969....307372602
..4..160...3958...70202....996083...11861187...122371876...1118328968
..7..232...6334..127954...2098908...29034355...348412793...3701885454
..4..254...8641..211261...4080930...65777365...914595808..11222027568
..6..364..13239..357465...7790731..142805210..2264326584..31699803817
..6..388..17240..549988..13869018..291148080..5259342780..83633494240
..9..542..25227..875942..24527424..576393509.11692554312.209004408969
..4..525..31128.1273587..40768032.1082580072.24683815072.494557127475
.10..767..44660.1941522..67955336.1993354411.50403390786
..6..754..53786.2705012.106952861.3505999065
..8.1015..74111.3966472.169909977
..6..993..86712.5325916
.12.1389.118779
..8.1256
.11

where the first column (=single (1..n-1) * (1..n-1) pair) is apparently
http://www.research.att.com/~njas/sequences/A007897

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