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
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).
Programs
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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
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Magma
m:=30; R
:=PowerSeriesRing(Integers(), m); Coefficients(R!( 1/((1-x^2)*(1-x-x^2)^2) )); // G. C. Greubel, Jan 31 2019 -
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 *) -
PARI
my(x='x+O('x^30)); Vec(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
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