A100504 a(n) = (4*n^3 + 6*n^2 + 8*n + 6)/3.

2, 8, 26, 64, 130, 232, 378, 576, 834, 1160, 1562, 2048, 2626, 3304, 4090, 4992, 6018, 7176, 8474, 9920, 11522, 13288, 15226, 17344, 19650, 22152, 24858, 27776, 30914, 34280, 37882, 41728, 45826, 50184, 54810, 59712, 64898, 70376, 76154, 82240

Offset: 0

Author

N. J. A. Sloane, Nov 24 2004

Keywords

• nonn
• easy

Comments

Bisection of A000125.

This sequence is related to A002061 by a(n) = (n+1)*A002061(n+1) + Sum_{i=0..n} A002061(i). - Bruno Berselli, Dec 19 2013

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A037237.

Programs

Magma

I:=[2, 8, 26, 64]; [n le 4 select I[n] else 4*Self(n-1) -6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Jun 26 2012

Mathematica

CoefficientList[Series[2*(1+3x^2)/((1-x)^4),{x,0,40}],x] (* Vincenzo Librandi, Jun 26 2012 *)
LinearRecurrence[{4,-6,4,-1},{2,8,26,64},40] (* Harvey P. Dale, Dec 27 2015 *)

PARI

a(n)=n*(4*n^2+6*n+8)/3+2 \\ Charles R Greathouse IV, Jan 18 2012

SageMath

[2 + 2*n*(2*n^2+3*n+4)/3 for n in range(41)] # G. C. Greubel, Apr 03 2023

Formula

a(n) = a(n-1) + (2*n)^2 + 2. - Philippe Deléham, Jan 18 2012

From Vincenzo Librandi, Jun 26 2012: (Start)

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

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). (End)

From G. C. Greubel, Apr 03 2023: (Start)

a(n) = 2 + 2*A037237(n-1).

E.g.f.: (2/3)*(3 + 9*x + 9*x^2 + 2*x^3)*exp(x). (End)

Extensions

More terms from Hugo Pfoertner, Nov 25 2004

New name based on formula from Ralf Stephan