A147600 Expansion of 1/(1 - 3*x^2 + x^4).

1, 0, 3, 0, 8, 0, 21, 0, 55, 0, 144, 0, 377, 0, 987, 0, 2584, 0, 6765, 0, 17711, 0, 46368, 0, 121393, 0, 317811, 0, 832040, 0, 2178309, 0, 5702887, 0, 14930352, 0, 39088169, 0, 102334155, 0, 267914296, 0, 701408733, 0, 1836311903, 0, 4807526976, 0

Offset: 0

Author

Roger L. Bagula, Nov 08 2008

Keywords

• nonn
• easy

Comments

S(n,sqrt(5)), with the Chebyshev polynomials A049310, is an integer sequence in the real quadratic number field Q(sqrt(5)) with basis numbers <1,phi>, phi:=(1+sqrt(5))/2. S(n,sqrt(5)) = A(n) + 2*B(n)*phi, with A(n) = A005013(n+1)*(-1)^n and B(n) = a(n-1), n>=0, with a(-1)=0. - Wolfdieter Lang, Nov 24 2010

The sequence (s(n)) given by s(0) = 0 and s(n) = a(n-1) for n > 0 is the p-INVERT of (0, 1, 0, 1, 0, 1, ...) using p(S) = 1 - S^2; see A291219. - Clark Kimberling, Aug 30 2017

From Jean-François Alcover, Sep 24 2017: (Start)

Consider this array of successive differences:

0, 0, 0, 1, 0, 3, 0, 8, 0, 21, ...

0, 0, 1, -1, 3, -3, 8, -8, 21, -21, ...

0, 1, -2, 4, -6, 11, -16, 29, -42, 76, ...

1, -3, 6, -10, 17, -27, 45, -71, 118, -186, ...

-4, 9, -16, 27, -44, 72, -116, 189, -304, 495, ...

13, -25, 43, -71, 116, -188, 305, -493, 799, -1291, ...

-38, 68, -114, 187, -304, 493, -798, 1292, -2090, 3383, ...

...

First row = even-index Fibonacci numbers with interleaved zeros = this sequence right-shifted 3 positions.

Main diagonal = 0,0,-2,-10,-44,-188,-798,... = -A099919 right-shifted.

First upper subdiagonal = 0,1,4,17,72,305,1292,... = A001076 right-shifted.

Second upper subdiagonal = 0,-1,-6,-27,-116,-493,-2090,... = -A049651.

Third upper subdiagonal = 1,3,11,45,189,799,3383,... = A292278.

(End) (Comment based on an e-mail from Paul Curtz)

Examples

			G.f. = 1 + 3*x^2 + 8*x^4 + 21*x^6 + 55*x^8 + 144*x^10 + 377*x^12 + 987*x^14 + ...
		

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (0,3,0,-1).

Crossrefs

Cf. A000045, A001076, A001906, A005013, A049310.

Cf. A049651, A088305, A099919, A291219, A292278.

Programs

Magma

[(1+(-1)^n)*Fibonacci(n+2)/2: n in [0..60]]; // G. C. Greubel, Oct 25 2022

Mathematica

f[x_]= -1 -x +x^2; CoefficientList[Series[-1/(x^2*f[x]*f[1/x]), {x,0,60}], x]
(* or *)
M={{0,1,0,0}, {0,0,1,0}, {0,0,0,1}, {-1,0,3,0}}; v[0]= {1,0,3,0}; v[n_]:= v[n]= M.v[n-1]; Table[v[n][[1]], {n,0,60}]
LinearRecurrence[{0,3,0,-1}, {1,0,3,0}, 60] (* Jean-François Alcover, Sep 23 2017 *)

PARI

Vec(1/(1 - 3*x^2 + x^4)+O(x^99)) \\ Charles R Greathouse IV, Sep 23 2012

SageMath

[((n+1)%2)*fibonacci(n+2) for n in range(60)] # G. C. Greubel, Oct 25 2022

Formula

O.g.f.: 1/(1 - 3*x^2 + x^4).

a(2*k) = F(2*(k+1)), a(2*k+1) = 0, k>=0, with F(n)=A000045(n). - Richard Choulet, Nov 13 2008

a(n) + a(n-1) + a(n-2) = A005013(n + 1). - Michael Somos, Apr 13 2012

a(n) = (2^(-2-n)*((1 + (-1)^n)*((-3+sqrt(5))*(-1+sqrt(5))^n + (1+sqrt(5))^n*(3+sqrt(5)))))/sqrt(5). - Colin Barker, Mar 28 2016