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A054454 Third column of triangle A054453. 6
1, 2, 6, 12, 26, 50, 97, 180, 332, 600, 1076, 1908, 3361, 5878, 10226, 17700, 30510, 52390, 89665, 153000, 260376, 442032, 748776, 1265832, 2136001, 3598250, 6052062, 10164540, 17048642 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (2, 2, -4, -2, 2, 1).

FORMULA

a(n) = A054453(n+2, 2).

a(2*k) = 1 + (8*n*Fibonacci(2*n+1) + 3*(2*n+1)*Fibonacci(2*n))/5.

a(2*k+1) = 2*(2*(2*n+1)*Fibonacci(2*(n+1)) + 3*(n+1)*Fibonacci(2*n+1))/5.

G.f.: ((Fib(x))^2)/(1-x^2), with Fib(x)=1/(1-x-x^2) = g.f. A000045(n+1)(Fibonacci numbers without F(0)).

a(n) = 2*a(n-1) + 2*a(n-2) - 4*a(n-3) - 2*a(n-4) + 2*a(n-5) + a(n-6) where a(0)=1, a(1)=2, a(2)=6, a(3)=12, a(4)=26, a(5)=50. -  Harvey P. Dale, May 06 2012

MATHEMATICA

CoefficientList[Series[(1/(1-x-x^2))^2/(1-x^2), {x, 0, 30}], x] (* or *) LinearRecurrence[{2, 2, -4, -2, 2, 1}, {1, 2, 6, 12, 26, 50}, 30] (* Harvey P. Dale, May 06 2012 *)

PROG

(PARI) my(x='x+O('x^30)); Vec(1/((1-x^2)*(1-x-x^2)^2)) \\ G. C. Greubel, Jan 31 2019

(MAGMA) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( 1/((1-x^2)*(1-x-x^2)^2) )); // G. C. Greubel, Jan 31 2019

(Sage) (1/((1-x^2)*(1-x-x^2)^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jan 31 2019

(GAP) a:=[1, 2, 6, 12, 26, 50];; for n in [7..30] do a[n]:=2*a[n-1]+2*a[n-2] -4*a[n-3]-2*a[n-4]+2*a[n-5]+a[n-6]; od; a; # G. C. Greubel, Jan 31 2019

CROSSREFS

Cf. A054453, A000045.

Sequence in context: A141347 A300120 A246584 * A084170 A245264 A327477

Adjacent sequences:  A054451 A054452 A054453 * A054455 A054456 A054457

KEYWORD

easy,nonn

AUTHOR

Wolfdieter Lang, Apr 27 2000

STATUS

approved

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Last modified May 31 16:29 EDT 2020. Contains 334748 sequences. (Running on oeis4.)