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 A142586 Binomial transform of A014217. 2
 1, 2, 5, 14, 39, 107, 290, 779, 2079, 5522, 14615, 38579, 101634, 267347, 702455, 1844114, 4838079, 12686507, 33254210, 87141659, 228301839, 598026002, 1566300455, 4101923939, 10741568514, 28126975907, 73647747815, 192833044754, 504884940879, 1321888886747 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS The second term in the k-th iterated differences is 2, 3, 6, 10, 17, 28, 46, ... = A001610(k+1). LINKS Colin Barker, Table of n, a(n) for n = 0..1000 Nickolas Hein and Jia Huang, Variations of the Catalan numbers from some nonassociative binary operations, arXiv:1807.04623 [math.CO], 2018. Index entries for linear recurrences with constant coefficients, signature (5,-7,2). FORMULA From R. J. Mathar, Sep 22 2008: (Start) G.f.: (1 - 3*x + 2*x^2 + x^3)/((1-3*x+x^2)*(1-2*x)). a(n) = A005248(n) - 2^(n-1), n>0. (End) a(n) = 5*a(n-1) - 7*a(n-2) + 2*a(n-3); a(0)=1, a(1)=2, a(2)=5, a(3)=14. - Harvey P. Dale, Aug 08 2011 a(n) = (-2^(-1+n) + ((3-sqrt(5))/2)^n + ((3+sqrt(5))/2)^n) for n > 0. - Colin Barker, Jun 05 2017 MAPLE 1, seq(combinat[fibonacci](2*n+1) +combinat[fibonacci](2*n-1) -2^(n-1), n = 1..30); # G. C. Greubel, Apr 13 2021 MATHEMATICA CoefficientList[Series[(1-3x+2x^2+x^3)/((1-3x+x^2)(1-2x)), {x, 0, 30}], x] (* or *) Join[{1}, LinearRecurrence[{5, -7, 2}, {2, 5, 14}, 30]] (* Harvey P. Dale, Aug 08 2011 *) PROG (PARI) Vec((1-3*x+2*x^2+x^3)/((1-3*x+x^2)*(1-2*x)) + O(x^30)) \\ Colin Barker, Jun 05 2017 (Magma) [1] cat [Lucas(2*n) - 2^(n-1): n in [1..30]]; // G. C. Greubel, Apr 13 2021 (Sage) [1]+[lucas_number2(2*n, 1, -1) -2^(n-1) for n in (1..30)] # G. C. Greubel, Apr 13 2021 CROSSREFS Cf. A001610, A005248. Sequence in context: A331573 A141752 A291729 * A202207 A132834 A000641 Adjacent sequences: A142583 A142584 A142585 * A142587 A142588 A142589 KEYWORD nonn,easy AUTHOR Paul Curtz, Sep 21 2008 EXTENSIONS Edited and extended by R. J. Mathar, Sep 22 2008 STATUS approved

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Last modified December 5 04:31 EST 2023. Contains 367574 sequences. (Running on oeis4.)