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A147600 Expansion of 1/(1 - 3*x^2 + x^4). 8
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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 zeroes = 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)

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).

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

EXAMPLE

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

MATHEMATICA

f[x_] = x^2-x-1; g[x] = ExpandAll[ -f[x]*x^2*f[1/x]]; a = Table[SeriesCoefficient[Series[1/g[x], {x, 0, 50}], n], {n, 0, 50}]

(* or: *)

M={{0, 1, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}, {-1, 0, 3, 0}}; v[0] = Table[a[[n]], {n, 1, 4}]={1, 0, 3, 0}; v[n_] := v[n] = M.v[n - 1]; Table[v[n][[1]], {n, 0, 50}]

(* or: *)

LinearRecurrence[{0, 3, 0, -1}, {1, 0, 3, 0}, 48] (* Jean-François Alcover, Sep 23 2017 *)

PROG

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

CROSSREFS

Cf. A000045, A001906, A005013, A088305.

Sequence in context: A209491 A201575 A194796 * A022895 A197416 A197512

Adjacent sequences:  A147597 A147598 A147599 * A147601 A147602 A147603

KEYWORD

nonn,easy

AUTHOR

Roger L. Bagula, Nov 08 2008

STATUS

approved

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Last modified September 28 09:32 EDT 2021. Contains 347714 sequences. (Running on oeis4.)