A096337 Number of those nonnegative integer solutions of the congruence x_1+2x_2+...+(n-1)x_{n-1} = 0 (mod n) which are indecomposable, that is, are not nonnegative linear combinations of other nonnegative integer solutions.

0, 1, 3, 6, 14, 19, 47, 64, 118, 165, 347, 366, 826, 973, 1493, 2134, 3912, 4037, 7935, 8246, 12966, 17475, 29161, 28064, 49608, 59357, 83419, 97242, 164966, 152547, 280351, 295290, 405918, 508161, 674629, 708818, 1230258, 1325731, 1709229, 1868564, 3045108

Offset

1,3

Comments

a(n) is a lower bound for the number of fundamental invariants of binary forms of degree n+2 - see Kac. A lower estimate for a(n) is given by Dixmier et al.

a(n) is the number of nonempty multisets of positive integers < n such that their sum modulo n is zero and that no proper nonempty subset has this property. - George B. Salomon, Sep 29 2019

Links

Vakhtang Tsiskaridze, Table of n, a(n) for n = 1..64, computed by a Pascal code (1994, unpublished)

J. Dixmier, P. Erdős and J.-L. Nicolas, Sur le nombre d'invariants fondamentaux des formes binaires, C. R. Acad. Sci. Paris Ser. I Math. 305 (1987), no. 8, 319-322.

John C. Harris and David L. Wehlau, Non-negative Integer Linear Congruences, Indag. Math. <b>17</b> (2006) 37-44.

V. Kac, Root systems, representations of quivers and invariant theory, Invariant theory (Montecatini, 1982), 74-108, Lecture Notes in Math., 996, Springer, Berlin, 1983.

Klaus Pommerening, The Indecomposable Solutions of Linear Congruences, arXiv:1703.03708 [math.NT], 2017.

Example

a(3)=3 since 3+2*0=3, 1+2*1=3 and 0+2*3=6 are the only indecomposable nonnegative integer solutions to x_1+2x_2=0 (mod 3): all other nonnegative integer solutions have form x_1=p*3+q*1+r*0, x_2=p*0+q*1+r*3 for nonnegative integers p, q, r.

Crossrefs

Sequence in context: A263620 A369304 A083356 * A175318 A281025 A109757

Adjacent sequences: A096334 A096335 A096336 * A096338 A096339 A096340

Keyword

nonn

Author

Mamuka Jibladze, Jun 28 2004

Status

approved