A381291 -id:A381291 - cp's OEIS frontend

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

Xavier Roulleau, Feb 21 2025

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.

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

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).

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

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

A381351 Number of subsets of 9 integers between 1 and n such that their sum is 3 modulo n.

Original entry on oeis.org

1, 5, 19, 55, 143, 335, 715, 1430, 2703, 4862, 8398, 14000, 22610, 35530, 54484, 81719, 120175, 173592, 246675, 345345, 476913, 650325, 876525, 1168710, 1542684, 2017356, 2615103, 3362260, 4289780, 5433736, 6835972, 8544965, 10616489, 13114465

Offset: 10

Xavier Roulleau, Feb 21 2025

For s an integer such that GCD(s,9)=3, this is also the number of subsets of 9 integers between 1 and n such that their sum is s modulo n.

			For n=10, there are a(10)=1 order 9 subsets of Z/10Z with sum equal to 3 mod 10.

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 (6,-15,22,-27,36,-42,36,-27,23,-21,21,-23,27,-36,42,-36,27,-22,15,-6,1).

Cf. A011796, A031164, A032194, A381289, A381290, A381291.

G.f.: x^10*(1 - x + 4*x^2 - 6*x^3 + 15*x^4 - 17*x^5 + 15*x^6 - 6*x^7 + 4*x^8 - x^9 + x^10)/((1 - x)^6*(1 - x^3)^2*(1 - x^9)).

cp's OEIS Frontend

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

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