A381350 Number of subsets of 8 integers between 1 and n such that their sum is 2 modulo n.

1, 5, 15, 42, 99, 217, 429, 808, 1430, 2438, 3978, 6308, 9690, 14550, 21318, 30664, 43263, 60115, 82225, 111038, 148005, 195143, 254475, 328752, 420732, 534076, 672452, 840648, 1043460, 1287036, 1577532, 1922736, 2330445, 2810385, 3372291, 4028178, 4790071, 5672645

Offset

9,2

Comments

For s an integer such that GCD(s,8)=2, this is also the number of subsets of 8 integers between 1 and n such that their sum is s modulo n.

Links

Stefano Spezia, Table of n, a(n) for n = 9..10000

David Broadhurst and Xavier Roulleau, Number of partitions of modular integers, arXiv:2502.19523 [math.NT], 2025.

Index entries for linear recurrences with constant coefficients, signature (4,-4,-4,12,-12,4,12,-22,12,4,-12,12,-4,-4,4,-1).

Formula

G.f.: x^9*(1 + x - x^2 + 6*x^3 + 2*x^5 + 6*x^7 - x^8 + x^9 + x^10)/((1 - x)^4*(1 - x^2)^2*(1 - x^4)*(1 - x^8)).

a(n) = (n - 4)*(2520 - 24*(281 + 35*(-1)^n)*n + 5*(1039 + 21*(-1)^n)*n^2 - 2112*n^3 + 452*n^4 - 48*n^5 + 2*n^6 - 2520*A056594(n))/80640. - Stefano Spezia, Feb 21 2025

Example

For n=10, there are a(10)=5 order 8 subsets of Z/10Z with sum equal to 2 mod 10.

Mathematica

Drop[CoefficientList[Series[x^9(1+x-x^2+6x^3+2x^5+6x^7-x^8+x^9+x^10)/((1-x)^4(1-x^2)^2(1-x^4)(1-x^8)), {x, 0, 50}], x], 9] (* Harvey P. Dale, Feb 22 2026 *)
(* Alternative: *)
LinearRecurrence[{4, -4, -4, 12, -12, 4, 12, -22, 12, 4, -12, 12, -4, -4, 4, -1}, {1, 5, 15, 42, 99, 217, 429, 808, 1430, 2438, 3978, 6308, 9690, 14550, 21318, 30664}, 40] (* Harvey P. Dale, Feb 22 2026 *)

Crossrefs

Cf. A011796, A031164, A056594, A381289, A381290, A381291.

Sequence in context: A102620 A211380 A053731 * A111295 A200760 A381291

Adjacent sequences: A381347 A381348 A381349 * A381351 A381352 A381353

Keyword

nonn,easy

Author

Xavier Roulleau, Feb 21 2025

Status

approved