A129370 a(n) = n^2 - (n-1)^2*(1 - (-1)^n)/8.

0, 1, 4, 8, 16, 21, 36, 40, 64, 65, 100, 96, 144, 133, 196, 176, 256, 225, 324, 280, 400, 341, 484, 408, 576, 481, 676, 560, 784, 645, 900, 736, 1024, 833, 1156, 936, 1296, 1045, 1444, 1160, 1600, 1281, 1764, 1408

Offset: 0

Author

Paul Barry, Apr 11 2007

Keywords

• easy
• nonn

Comments

Partial sums are A129371.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000
• Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Crossrefs

Cf. A000567 (odd bisection), A016742 (even bisection), A129371.

Programs

Magma

[n^2 -(n-1)^2*(n mod 2)/4: n in [0..60]]; // G. C. Greubel, Jan 31 2024

Mathematica

Table[n^2-(n-1)^2 (1-(-1)^n)/8,{n,0,50}] (* Harvey P. Dale, Oct 22 2011 *)

PARI

a(n)=n^2-(n-1)^2*(1-(-1)^n)/8 \\ Charles R Greathouse IV, Sep 28 2015

SageMath

[n^2 -(n-1)^2*(n%2)/4 for n in range(61)] # G. C. Greubel, Jan 31 2024

Formula

a(n) = (1/8)*( (7*n^2 + 2*n - 1) + (-1)^n*(n-1)^2 ).

G.f.: x*(1 + 4*x + 5*x^2 + 4*x^3)/(1-x^2)^3.

E.g.f.: (1/4)*( x*(5+4*x)*cosh(x) - (1-4*x-3*x^2)*sinh(x) ). - G. C. Greubel, Jan 31 2024