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

0,1

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).

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)

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 *)

Prog

(PARI) a(n)=n*(4*n^2+6*n+8)/3+2 \\ Charles R Greathouse IV, Jan 18 2012
(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
(SageMath) [2 + 2*n*(2*n^2+3*n+4)/3 for n in range(41)] # G. C. Greubel, Apr 03 2023

Crossrefs

Cf. A037237.

Sequence in context: A212527 A227015 A216929 * A327600 A099416 A211885

Adjacent sequences: A100501 A100502 A100503 * A100505 A100506 A100507

Keyword

nonn,easy

Author

N. J. A. Sloane, Nov 24 2004

Extensions

More terms from Hugo Pfoertner, Nov 25 2004

New name based on formula from Ralf Stephan

Status

approved