A142710 a(n) = A142585(n) + A142586(n).

2, 2, 6, 14, 38, 112, 276, 814, 1998, 5702, 14226, 39404, 99908, 270922, 695106, 1859134, 4807518, 12748472, 33128916, 87394454, 227792678, 599050102, 1564242906, 4106054164, 10733283588, 28143585362, 73614464826, 192899714414, 504751433798, 1322156172352

Offset: 0

Author

Paul Curtz, Sep 25 2008

Keywords

• nonn
• easy

Comments

Sum of the binomial and inverse binomial transforms of A014217.

Starting at a(1), the last digits form a period-4 sequence 2, 6, 4, 8.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,7,-12,-11,16,-4).

Crossrefs

Cf. A000032, A014217, A142585, A142586, A147586.

Programs

Magma

[n eq 0 select 2 else (-1)^n*Lucas(n) +Lucas(2*n) -(1+(-1)^n)*2^(n-1): n in [0..50]]; // G. C. Greubel, Oct 26 2022

Mathematica

Join[{2},LinearRecurrence[{2,7,-12,-11,16,-4},{2,6,14,38,112,276},30]] (* Harvey P. Dale, Nov 25 2013 *)

SageMath

def A142710(n): return (-1)^n*lucas_number2(n,1,-1) + lucas_number2(2*n,1,-1) - (1 + (-1)^n)*2^(n-1) -int(n==0)
[A142710(n) for n in range(51)] # G. C. Greubel, Oct 26 2022

Formula

a(n) = +2*a(n-1) +7*a(n-2) -12*a(n-3) -11*a(n-4) +16*a(n-5) -4*a(n-6), n>6. - R. J. Mathar, Jun 14 2010

G.f.: 2*(1-x-6*x^2+6*x^3+7*x^4-2*x^6)/((1-2*x)*(1+2*x)*(1+x-x^2)*(1-3*x+x^2)). - Colin Barker, Aug 13 2012

a(n) = (-1)^n*LucasL(n) + LucasL(2*n) - (1 + (-1)^n)*2^(n-1) - [n=0]. - G. C. Greubel, Oct 26 2022

Extensions

Offset set to zero and extended - R. J. Mathar, Jun 14 2010