A330133 a(n) = (1/16)*(5 + (-1)^(1+n) - 4*cos(n*Pi/2) + 10*n^2).

0, 1, 3, 6, 10, 16, 23, 31, 40, 51, 63, 76, 90, 106, 123, 141, 160, 181, 203, 226, 250, 276, 303, 331, 360, 391, 423, 456, 490, 526, 563, 601, 640, 681, 723, 766, 810, 856, 903, 951, 1000, 1051, 1103, 1156, 1210, 1266, 1323, 1381, 1440, 1501, 1563, 1626, 1690, 1756

Offset

0,3

Comments

For n > 0, partial sums of A047201.

Links

Table of n, a(n) for n=0..53.

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

Formula

O.g.f.: -x*(1 + x + x^2 + x^3 + x^4)/((-1 + x)^3*(1 + x)*(1 + x^2)).

E.g.f.: (1/16)*(-exp(-x) + 5*exp(x)*(1 + 2*x + x^2) - 4*cos(x)).

a(n) = 2*a(n-1) - a(n-2) + a(n-4) -2*a(n-5) + a(n-6) for n > 5.

a(2*n-1) = A005891(n-1) for n > 0.

a(4*n) = 10*n^2. - Bernard Schott, Dec 06 2019

Maple

gf:=(1/16)*(-exp(-x) + 5*exp(x)*(1 + 2*x + 2*x^2) - 4*cos(x)); ser := series(gf, x, 54):
seq(factorial(n)*coeff(ser, x, n), n = 0 .. 53)

Mathematica

Table[(1/16)*(5+(-1)^(1+n)-4*Cos[n*Pi/2]+10*n^2), {n, 0, 53}]
LinearRecurrence[{2, -1, 0, 1, -2, 1}, {0, 1, 3, 6, 10, 16}, 60] (* Harvey P. Dale, Jul 21 2021 *)

Prog

(Magma) I:=[0, 1, 3, 6, 10, 16]; [n le 6 select I[n] else 2*Self(n-1)-Self(n-2)+Self(n-4)-2*Self(n-5)+Self(n-6): n in [1..54]];
(PARI) concat([0], Vec(-x*(1 + x + x^2 + x^3 + x^4)/((-1 + x)^3*(1 + x)*(1 + x^2))+O(x^54)))

Crossrefs

Cf. A005891, A033583 (10*n^2), A047201.

Sequence in context: A111712 A293407 A194144 * A025701 A025744 A180746

Adjacent sequences: A330130 A330131 A330132 * A330134 A330135 A330136

Keyword

easy,nonn

Author

Stefano Spezia, Dec 02 2019

Status

approved