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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 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. 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. 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:=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

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Last modified June 10 20:56 EDT 2023. Contains 363207 sequences. (Running on oeis4.)