A114220 a(n) = Sum_{k=0..floor(n/2)} (k - (k-1)*0^(n-2*k)).

1, 0, 1, 1, 2, 3, 4, 6, 7, 10, 11, 15, 16, 21, 22, 28, 29, 36, 37, 45, 46, 55, 56, 66, 67, 78, 79, 91, 92, 105, 106, 120, 121, 136, 137, 153, 154, 171, 172, 190, 191, 210, 211, 231, 232, 253, 254, 276, 277, 300, 301, 325, 326, 351, 352, 378, 379, 406, 407, 435, 436

Offset: 0

Author

Paul Barry, Nov 18 2005

Keywords

• easy
• nonn

Comments

Diagonal sums of A114219.

It appears that the sequence terms from a(4) = 2 onwards are the exponents in the expansion of Sum_{n >= 0} q^(3*n-3) * Product_{k >= n} 1 - q^(2*k) = 1 - q^2 + q^3 - q^4 + q^6 - q^7 + q^10 - q^11 + - .... - Peter Bala, Jan 17 2025

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1)

Crossrefs

Cf. A055802, A114219.

Column k=2 of A309049 (for n>0).

Programs

Magma

[(2*n^2-2*n+7 + (9-2*n)*(-1)^n)/16: n in [0..80]]; // G. C. Greubel, Oct 21 2024

Mathematica

CoefficientList[Series[(1-x-x^2+2x^3)/((1-x)(1-x^2)^2), {x,0,80}],x] (* Harvey P. Dale, Mar 24 2011 *)

SageMath

def A114220(n): return (2*n^2-2*n+7 + (9-2*n)*(-1)^n)//16
[A114220(n) for n in range(81)] # G. C. Greubel, Oct 21 2024

Formula

G.f.: (1-x-x^2+2x^3)/((1-x)*(1-x^2)^2).

a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).

a(n) = (2*n^2-2*n+7 + (9-2*n)*(-1)^n)/16.

a(n) = A055802(n+1), n > 1. - R. J. Mathar, Aug 11 2008

E.g.f.: (1/16)*((9 + 2*x)*exp(-x) + (7 + 2*x^2)*exp(x)). - G. C. Greubel, Oct 21 2024