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.

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, 25, 24, 28, 26, 30, 26, 31, 32, 34, 28, 36, 36, 37, 34, 40, 38, 42, 40, 43, 44, 46, 34, 48, 48, 49, 48, 52, 50, 54, 50, 55, 56, 58, 52, 60, 60

Offset: 0

Author

Brooke Logan, Jun 25 2016

Keywords

• nonn

Links

• Brooke Logan, Table of n, a(n) for n = 0..10000

Crossrefs

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.

Formula

We argue as in A273724. There are n-1 choices for i.

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.

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.

Putting the pieces together, we obtain the g.f.

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.

The g.f. can also be written as

(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)).