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

Author

R. J. Mathar, Nov 08 2008

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A040000, A042964, A049347, A071619.

Programs

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

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 *)

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

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)