cp's OEIS Frontend

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.

%I A096337 #28 Jun 23 2024 20:00:22
%S A096337 0,1,3,6,14,19,47,64,118,165,347,366,826,973,1493,2134,3912,4037,7935,
%T A096337 8246,12966,17475,29161,28064,49608,59357,83419,97242,164966,152547,
%U A096337 280351,295290,405918,508161,674629,708818,1230258,1325731,1709229,1868564,3045108
%N 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.
%C A096337 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.
%C A096337 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
%H A096337 Vakhtang Tsiskaridze, <a href="/A096337/b096337.txt">Table of n, a(n) for n = 1..64</a>, computed by a Pascal code (1994, unpublished)
%H A096337 J. Dixmier, P. Erdős and J.-L. Nicolas, <a href="http://gallica.bnf.fr/ark:/12148/bpt6k5744571t/f9.image">Sur le nombre d'invariants fondamentaux des formes binaires</a>, C. R. Acad. Sci. Paris Ser. I Math. 305 (1987), no. 8, 319-322.
%H A096337 John C. Harris and David L. Wehlau, <a href="https://doi.org/10.1016/S0019-3577(06)80005-4">Non-negative Integer Linear Congruences</a>, Indag. Math. 17 (2006) 37-44.
%H A096337 V. Kac, <a href="http://dx.doi.org/10.1007/BFb0063236">Root systems, representations of quivers and invariant theory</a>, Invariant theory (Montecatini, 1982), 74-108, Lecture Notes in Math., 996, Springer, Berlin, 1983.
%H A096337 Klaus Pommerening, <a href="https://arxiv.org/abs/1703.03708">The Indecomposable Solutions of Linear Congruences</a>, arXiv:1703.03708 [math.NT], 2017.
%e A096337 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.
%K A096337 nonn
%O A096337 1,3
%A A096337 _Mamuka Jibladze_, Jun 28 2004