A054454 Third column of triangle A054453.

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

Offset

0,2

Links

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

Charles H. Conley and Valentin Ovsienko, Shadows of rationals and irrationals: supersymmetric continued fractions and the super modular group, arXiv:2209.10426 [math-ph], 2022.

Kálmán Liptai, László Németh, Tamás Szakács, and László Szalay, On certain Fibonacci representations, arXiv:2403.15053 [math.NT], 2024. See p. 8.

Gregg Musiker, Nick Ovenhouse, and Sylvester W. Zhang, Double Dimers and Super Ptolemy Relations, Séminaire Lotharingien de Combinatoire XX, Proc. 35th Conf. Formal Power, Series and Algebraic Combinatorics (Davis) 2023, Art. #YY. See p. 12.

László Németh, Walks on tiled boards, arXiv:2403.12159 [math.CO], 2024. See p. 3.

Tamás Szakács, Linear recursive sequences and factorials, Ph. D. Thesis, Univ. Debrecen (Hungary, 2024). See p. 23.

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: A335724 A300120 A246584 * A084170 A245264 A327477

Adjacent sequences: A054451 A054452 A054453 * A054455 A054456 A054457

Keyword

easy,nonn

Author

Wolfdieter Lang, Apr 27 2000

Status

approved