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A183954 Number of strings of numbers x(i=1..3) in 0..n with sum i^2*x(i) equal to n*9.

%I A183954 #13 Jul 22 2022 05:14:58
%S A183954 1,2,2,3,4,6,7,9,12,14,17,19,22,25,29,32,36,41,45,50,54,59,64,70,75,
%T A183954 81,88,94,101,107,114,121,129,136,144,153,161,170,178,187,196,206,215,
%U A183954 225,236,246,257,267,278,289,301,312,324,337,349,362,374,387,400,414,427,441
%N A183954 Number of strings of numbers x(i=1..3) in 0..n with sum i^2*x(i) equal to n*9.
%C A183954 Row 3 of A183953.
%H A183954 R. H. Hardin, <a href="/A183954/b183954.txt">Table of n, a(n) for n = 1..200</a>
%F A183954 Empirical: a(n) = 2*a(n-1) - a(n-2) + a(n-9) - 2*a(n-10) + a(n-11).
%F A183954 Empirical g.f.: x*(1 + x)*(1 - x + x^3 - x^4 + 2*x^5 - 3*x^6 + 4*x^7 - 3*x^8 + x^9) / ((1 - x)^3*(1 + x + x^2)*(1 + x^3 + x^6)). - _Colin Barker_, Apr 07 2018
%e A183954 All solutions for n=3:
%e A183954 ..0....1
%e A183954 ..0....2
%e A183954 ..3....2
%t A183954 r[n_, k_, s_] := r[n, k, s] = Which[s < 0, 0, n == 0, If[s == 0, 1, 0], True, Sum[r[n - 1, k, s - c*n^2], {c, 0, k}]];
%t A183954 T[n_, k_] := r[n, k, k*n^2];
%t A183954 a[n_] := T[3, n];
%t A183954 Table[a[n], {n, 1, 62}] (* _Jean-François Alcover_, Jul 22 2022, after _R. J. Mathar_ in A183953 *)
%Y A183954 Cf. A183953.
%K A183954 nonn
%O A183954 1,2
%A A183954 _R. H. Hardin_, Jan 08 2011