A377166 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).

0, 21, 144, 273, 277, 448, 817, 1096, 1104, 1425, 2040, 2469, 2481, 2952, 3813, 4392, 4408, 5029, 6136, 6865, 6885, 7656, 9009, 9888, 9912, 10833, 12432, 13461, 13489, 14560, 16405, 17584, 17616, 18837, 20928, 22257, 22293, 23664, 26001, 27480, 27520, 29041, 31624, 33253, 33297, 34968

Offset: 0

Author

Stefano Spezia, Oct 18 2024

Keywords

• nonn
• easy

Comments

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.

Links

• Index entries for linear recurrences with constant coefficients, signature (1,0,0,2,-2,0,0,-1,1).

Crossrefs

Cf. A000290, A056594, A377165.

Programs

Mathematica

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]

Formula

a(n) = a(n-1) + 2*a(n-4) - 2*a(n-5) - a(n-8) + a(n-9) for n > 8.

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.

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.