A176281 Hankel transform of A176280.

1, 3, 12, 56, 280, 1440, 7488, 39104, 204544, 1070592, 5604864, 29345792, 153653248, 804532224, 4212572160, 22057287680, 115493404672, 604731211776, 3166413520896, 16579556016128, 86811681488896, 454551863820288

Offset: 0

Author

Paul Barry, Apr 14 2010

Keywords

• easy
• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (8,-16,8).

Crossrefs

Cf. A000045, A082761, A176280.

Programs

GAP

List([0..30], n-> 2^(n-1)*(Fibonacci(2*n+1) + 1)); # G. C. Greubel, Nov 24 2019

Magma

[2^(n-1)*(Fibonacci(2*n+1) + 1): n in [0..30]]; // G. C. Greubel, Nov 24 2019

Maple

with(combinat); seq(2^(n-1)*(fibonacci(2*n+1) + 1), n=0..30); # G. C. Greubel, Nov 24 2019

Mathematica

CoefficientList[Series[(1-5x+4x^2)/((1-2x)(1-6x+4x^2)),{x,0,40}],x] (* or *) LinearRecurrence[{8,-16,8},{1,3,12},40] (* Harvey P. Dale, Aug 14 2013 *)

PARI

vector(31, n, 2^(n-2)*(fibonacci(2*n-1) + 1)) \\ G. C. Greubel, Nov 24 2019

Sage

[2^(n-1)*(fibonacci(2*n+1) + 1) for n in (0..30)] # G. C. Greubel, Nov 24 2019

Formula

G.f.: (1-5*x+4*x^2)/(1-8*x+16*x^2-8*x^3) = (1-5*x+4*x^2)/((1-2*x)*(1-6*x+4*x^2)).

a(n) = 2^(n-1) + (3-sqrt(5))^n*((5-sqrt(5))/20) + (3+sqrt(5))^n*((5+sqrt(5))/20).

a(n) = 2^(n-1) + A082761(n)/2. - R. J. Mathar, Sep 30 2012

a(0)=1, a(1)=3, a(2)=12, a(n) = 8*a(n-1) - 16*a(n-2) + 8*a(n-3). - Harvey P. Dale, Aug 14 2013

a(n) = 2^(n-1)*(Fibonacci(2*n+1) + 1). - G. C. Greubel, Nov 24 2019