%I #10 Oct 18 2024 18:09:08
%S 0,21,144,273,277,448,817,1096,1104,1425,2040,2469,2481,2952,3813,
%T 4392,4408,5029,6136,6865,6885,7656,9009,9888,9912,10833,12432,13461,
%U 13489,14560,16405,17584,17616,18837,20928,22257,22293,23664,26001,27480,27520,29041,31624,33253,33297,34968
%N Expansion of g.f. x*(21 + 123*x + 129*x^2 + 4*x^3 + 129*x^4 + 123*x^5 + 21*x^6)/((1 - x)^3*(1 + x + x^2 + x^3)^2).
%C Numbers k such that 275*k + 1 is a square. The set of the integer square roots of 275*k + 1 is a superset of A377165.
%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,2,-2,0,0,-1,1).
%F a(n) = a(n-1) + 2*a(n-4) - a(n-8) + a(n-9) for n > 8.
%F a(n) = (245 + 550*n*(1 + n) - 25*(-1)^n*(1 + 2*n) - 44*(5 + 11*n)*A056594(n) - 44*(6 + 11*n)*A056594(n-1))/32.
%F E.g.f.: (5*(22 + 115*x + 55*x^2)*cosh(x) - 22*((5 + 11*x)*cos(x) + (6 - 11*x)*sin(x)) + 5*(27 + 105*x + 55*x^2)*sinh(x))/16.
%t CoefficientList[Series[x*(21 + 123*x + 129*x^2 + 4*x^3 + 129*x^4 + 123*x^5 + 21*x^6)/((1 - x)^3*(1 + x + x^2 + x^3)^2),{x,0,45}],x]
%Y Cf. A000290, A056594, A377165.
%K nonn,easy
%O 0,2
%A _Stefano Spezia_, Oct 18 2024