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

Ron Hardin rhhardin at att.net
Mon Sep 6 03:10:26 CEST 2010

```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

0,0,0,1,0,2,0,3,3,4,0,9,0,6,8,10,0,15,0,17,12,10,0,27,10,12,15,25,0,38,
0,26,20,16,24,51,0,18,24,51,0,56,0,41,51,22,0,74,21,50,32,49,0,69,40,75,
36,28,0,121,0,30,75,68,48,92,0,65,44,106,0,141,0,36,90,73,60,110,0,138
Number of distinct no-zero solutions of sum{i=1..1}(x(2i-1)*x(2i)) = 0 (mod n)
<a href="b000001.txt">Table of n, a(n) for n=1..152</a>

0,1,2,6,13,23,36,62,78,123,150,238,255,355,427,567,580,864,810,1202,
1232,1471,1452,2310,1960,2479,2712,3414,2947,4649,3600,5126,5022,5673,
5845,8457,6165,7975,8405,11062,8410,13133,9702,13726,14475,14323,12696
Number of distinct no-zero solutions of sum{i=1..2}(x(2i-1)*x(2i)) = 0 (mod n)
<a href="b000002.txt">Table of n, a(n) for n=1..445</a>

0,0,5,14,42,119,253,526,944,1702,2655,4510,6284,9867,13519,19980,25128,
37286,44562,63802,76850,102370,118591,168208,183320,241554,279852,
359754,387016,528861,543620,712450,780810,953120,1025355,1360255
Number of distinct no-zero solutions of sum{i=1..3}(x(2i-1)*x(2i)) = 0 (mod n)
<a href="b000003.txt">Table of n, a(n) for n=1..108</a>

0,1,3,34,145,522,1518,4041,9150,19970,38555,74370,128040,224434,358988,
587014,876114,1372578,1941624,2912816,4001868,5742391,7599933,10831065,
13788935,18946564,24080514,32270596,39619720,53256875,63655605,83580675
Number of distinct no-zero solutions of sum{i=1..4}(x(2i-1)*x(2i)) = 0 (mod n)
<a href="b000004.txt">Table of n, a(n) for n=1..50</a>

0,0,7,62,402,1960,7587,25470,73146,193036,455161,1022770,2098782,
4196208,7797217,14286908,24527232,42125055,67955391,110571248,170260213,
264585538,390623343,588725404,837466268,1220597598,1696535256
Number of distinct no-zero solutions of sum{i=1..5}(x(2i-1)*x(2i)) = 0 (mod n)
<a href="b000005.txt">Table of n, a(n) for n=1..34</a>

0,1,12,118,995,6518,32893,139369,499659,1599064,4551225,12014040,
29034510,66658219,142828285,295240250,576393604,1098913338,1993354080,
3560486370,6091808112,10312711744,16796812703,27247744649,42564714900
Number of distinct no-zero solutions of sum{i=1..6}(x(2i-1)*x(2i)) = 0 (mod n)
<a href="b000006.txt">Table of n, a(n) for n=1..30</a>

0,0,9,198,2294,19467,126864,674922,2997717,11615754,39660995,123162204,
348412638,920581314,2264397902,5294830516,11692554408,24858751383,
50403391227,99282023930,187902783369,347689331443,621482071507
Number of distinct no-zero solutions of sum{i=1..7}(x(2i-1)*x(2i)) = 0 (mod n)
<a href="b000007.txt">Table of n, a(n) for n=1..32</a>

0,1,15,327,4856,53455,444003,2948305,16112289,75389116,307372600,
1122069080,3701885580,11258893954,31699979961,83910860201,209004408715,
496246703439,1121475446118,2440154664350,5096346969372,10323359668079
Number of distinct no-zero solutions of sum{i=1..8}(x(2i-1)*x(2i)) = 0 (mod n)
<a href="b000008.txt">Table of n, a(n) for n=1..24</a>

0,0,22,499,9730,136491,1430727,11783122,78770456,443607864,2151608155,
9218591346,35373572400,123749340262,398005623516,1192411118090,
3344070542568,8869510553867,22304900540593,53635016669434
Number of distinct no-zero solutions of sum{i=1..9}(x(2i-1)*x(2i)) = 0 (mod n)
<a href="b000009.txt">Table of n, a(n) for n=1..30</a>

0,0,0,0,1,0,0,0,2,1,0,1,5,6,0,0,0,3,14,13,2,0,1,7,34,42,23,0,0,0,12,62,
145,119,36,3,0,1,9,118,402,522,253,62,3,0,0,15,198,995,1960,1518,526,78,
4,0,1,22,327,2294,6518,7587,4041,944,123,0,0,0,18,499,4856,19467,32893
T(n,k)=number of distinct no-zero solutions of sum{i=1..k}(x(2i-1)*x(2i)) = 0
(mod n)
<a href="b000010.txt">Table of n, a(n) for n=1..337</a>

rhhardin at mindspring.com
rhhardin at att.net (either)

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