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A274516 Place n equally-spaced points around a circle, labeled 0,1,2,...,n-1. For each i = 0..n-1 such that 5i != i mod n, draw an (undirected) chord from i to (5i mod n). Then a(n) is the total number of distinct chords.

%I A274516 #8 Jun 25 2016 14:55:08
%S A274516 0,0,0,1,0,4,2,6,2,7,8,10,4,12,12,13,10,16,14,18,16,19,20,22,10,24,24,
%T A274516 25,24,28,26,30,26,31,32,34,28,36,36,37,34,40,38,42,40,43,44,46,34,48,
%U A274516 48,49,48,52,50,54,50,55,56,58,52,60,60
%N A274516 Place n equally-spaced points around a circle, labeled 0,1,2,...,n-1. For each i = 0..n-1 such that 5i != i mod n, draw an (undirected) chord from i to (5i mod n). Then a(n) is the total number of distinct chords.
%H A274516 Brooke Logan, <a href="/A274516/b274516.txt">Table of n, a(n) for n = 0..10000</a>
%F A274516 We argue as in A273724. There are n-1 choices for i.
%F A274516 For nontrivial chords we need i != 5i mod n, which means 4i != 0 mod n, and so when n == 0 mod 4 we must subtract 3 from n-1 and when n == 2 mod 4 we must subtract 1 from n-1.
%F A274516 A chord occurs twice (but must be counted only once) when j==5i mod n and i==5j mod n, thus when 24i == 0 mod n. If n == +/- 3, +/- 9 mod 24 then subtract another 1, if n == +/- 6, +/- 8 mod 24 then subtract another 2, if n==12 mod 24 subtract 4, and if n == 0 mod 24 then subtract another 10.
%F A274516 Putting the pieces together, we obtain the g.f.
%F A274516 x^2/(1-x)^2-(3+x^2)/(1-x^4)-(x^3+x^9+x^15+x^21)/(1-x^24)-2(x^6+x^8+x^16+x^18)/(1-x^24)-(4*x^12+10)/(1-x^24)+13.
%F A274516 The g.f. can also be written as
%F A274516 (14*x^25 - 12*x^24 + 2*x^23 + x^22 + 3*x^21 - 2*x^20 + 2*x^19 + 4*x^17 - x^16 + x^15 + 8*x^13 - 6*x^12 + 2*x^11 + x^10 + 3*x^9 - 2*x^8 + 2*x^7 + 4*x^5 - x^4 + x^3 + 2*x - 2) / ((1-x)*(1-x^24)).
%Y A274516 If 5i in the definition is replaced by 2i we get A117571, if 5i is replaced by 3i we get A273724, and if 5i is replaced by 4i we get A274462.
%K A274516 nonn
%O A274516 0,6
%A A274516 _Brooke Logan_, Jun 25 2016