A147623 The 3rd Witt transform of A040000.

0, 2, 6, 12, 22, 34, 48, 66, 86, 108, 134, 162, 192, 226, 262, 300, 342, 386, 432, 482, 534, 588, 646, 706, 768, 834, 902, 972, 1046, 1122, 1200, 1282, 1366, 1452, 1542, 1634, 1728, 1826, 1926, 2028, 2134, 2242, 2352, 2466, 2582, 2700, 2822, 2946, 3072

Offset

0,2

Comments

The 2nd Witt transform of A040000 is represented by A042964.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Pieter Moree, The formal series Witt transform, Discr. Math. no. 295 vol. 1-3 (2005) 143-160.

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

Formula

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

a(n) = 2*A071619(n).

From G. C. Greubel, Oct 24 2022: (Start)

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

a(n) = 2*(2*(1 + 3*n^2) - (2*A049347(n) + A049347(n-1)))/9. (End)

Mathematica

CoefficientList[Series[2x(1+x)(1 +x^2)/((1-x)^3 (1+x+x^2)), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 14 2012 *)
LinearRecurrence[{2, -1, 1, -2, 1}, {0, 2, 6, 12, 22}, 50] (* Harvey P. Dale, Jul 04 2021 *)

Prog

(Magma) [n le 2 select 1+(-1)^n else 4*(1+(n-2)^2) - Self(n-1) - Self(n-2): n in [1..30]]; // G. C. Greubel, Oct 24 2022
(SageMath) [2*(2*(1+3*n^2) -(2*chebyshev_U(n, -1/2) +chebyshev_U(n-1, -1/2)))/9 for n in range(41)] # G. C. Greubel, Oct 24 2022

Crossrefs

Cf. A040000, A042964, A049347, A071619.

Sequence in context: A046960 A126428 A187490 * A232234 A045964 A005819

Adjacent sequences: A147620 A147621 A147622 * A147624 A147625 A147626

Keyword

nonn,easy

Author

R. J. Mathar, Nov 08 2008

Status

approved