%I #21 Nov 08 2024 15:33:22
%S 1,1,4,12,30,65,127,225,374,588,884,1281,1801,2465,3300,4332,5590,
%T 7105,8911,11041,13534,16428,19764,23585,27937,32865,38420,44652,
%U 51614,59361,67951,77441,87894,99372,111940,125665,140617,156865,174484,193548,214134
%N Number of partitions of n^2 into at most three parts.
%H Colin Barker, <a href="/A274250/b274250.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,-1,0,1,2,-3,1).
%F Coefficient of x^(n^2) in 1/((1-x)*(1-x^2)*(1-x^3)).
%F G.f.: (1-2*x+3*x^2+3*x^3+3*x^4+2*x^5+x^6+x^7) / ((1-x)^5*(1+x)*(1+x+x^2)).
%F a(n) = A001399(n^2) = round((n^2+3)^2/12). - _Alois P. Heinz_, Jun 16 2016
%t LinearRecurrence[{3,-2,-1,0,1,2,-3,1},{1,1,4,12,30,65,127,225},50] (* _Harvey P. Dale_, Nov 08 2024 *)
%o (PARI)
%o \\ b(n) is the coefficient of x^n in the g.f. 1/((1-x)*(1-x^2)*(1-x^3)).
%o b(n) = round(real((47+9*(-1)^n + 8*exp(-2/3*I*n*Pi) + 8*exp((2*I*n*Pi)/3) + 36*n+6*n^2)/72))
%o vector(50, n, n--; b(n^2))
%Y A subsequence of A001399.
%Y Cf. A274251 (n^3), A274252 (n^5), A274253 (n^7), A274254 (n^11).
%K nonn,easy,changed
%O 0,3
%A _Colin Barker_, Jun 16 2016