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 A083859 Main diagonal of generalized Fibonacci array A083856. 6
 0, 1, 1, 4, 9, 41, 133, 673, 2737, 15130, 72181, 430739, 2320825, 14815529, 88005541, 596681296, 3843559137, 27515587661, 189933449365, 1428716457761, 10474213334761, 82448447397646, 637534807917701, 5233087759204967, 42445677865505425, 362213650380301201 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS If a sequence (s(n): n >= 0) is of the form s(0) = 0, s(1) = x, and s(n) = s(n-1) + k*s(n-2) for n >= 2 (for some integer k >= 0 and some number x), then s(k) = a(k)*x. For example if k = 7 and x = 5, then (s(n): n = 0..7) = (0, 5, 5, 40, 75, 355, 880, 3365) and s(7) = 3365 = 673*5 = a(7)*x. - Gary Detlefs, Dec 04 2009 [Edited by Petros Hadjicostas, Dec 24 2019] LINKS G. C. Greubel, Table of n, a(n) for n = 0..690 FORMULA a(n) = (((1 + sqrt(4*n + 1))/2)^n - ((1 - sqrt(4*n + 1))/2)^n)/sqrt(4*n + 1). a(n) = A193376(n-1,n) for n >= 2. - R. J. Mathar, Aug 23 2011 a(n) = y(n,n), where y(m+2,n) = y(m+1,n) + n*y(m,n) with y(0,n) = 0 and y(1,n) = 1 for all n. - Benedict W. J. Irwin, Nov 03 2016 a(n) = [x^n] x/(1 - x - n*x^2). - Ilya Gutkovskiy, Oct 10 2017 a(n) = Sum_{s = 0..floor((n-1)/2)} binomial(n-1-s, s) * n^s. - Petros Hadjicostas, Dec 24 2019 From G. C. Greubel, Dec 27 2019: (Start) a(n) = (sqrt(n))^n * Fibonacci(n, 1/sqrt(n)), with a(0)=0. a(n) = (-sqrt(n)*i)^(n-1)*ChebyshevU(n-1, i/(2*sqrt(n))), with a(0)=0. (End) MAPLE seq( `if`(n=0, 0, simplify( (-sqrt(n)*I)^(n-1)*ChebyshevU(n-1, I/(2*sqrt(n)))) ), n=0..30); # G. C. Greubel, Dec 27 2019 # second Maple program: a:= n-> (<<0|1>, >^n)[1, 2]: seq(a(n), n=0..25); # Alois P. Heinz, Oct 19 2021 MATHEMATICA Table[DifferenceRoot[Function[{y, m}, {y[2 + m] == y[1 + m] + n*y[m], y[0] == 0, y[1] == 1}]][n], {n, 0, 20}] (* Benedict W. J. Irwin, Nov 03 2016 *) Table[If[n==0, 0, Round[(Sqrt[n])^(n-1)*Fibonacci[n, 1/Sqrt[n]] ]], {n, 0, 30}] (* G. C. Greubel, Dec 27 2019 *) PROG (PARI) vector(31, n, if(n==1, 0, round((-sqrt(n-1)*I)^(n-2)*polchebyshev(n-2, 2, I/(2*sqrt(n-1)))) ) ) \\ G. C. Greubel, Dec 27 2019 (Magma) [0] cat [ &+[Binomial(n-j-1, j)*n^j: j in [0..Floor((n-1)/2)]] : n in [1..30]]; // G. C. Greubel, Dec 27 2019 (Sage) [0]+[(-sqrt(n)*I)^(n-1)*chebyshev_U(n-1, I/(2*sqrt(n))) for n in (1..30)] # G. C. Greubel, Dec 27 2019 (GAP) Concatenation([0], List([1..30], n-> Sum([0..Int((n-1)/2)], j-> Binomial(n-j-1, j)*n^j) )); # G. C. Greubel, Dec 27 2019 CROSSREFS Cf. A083856, A083860, A193376. Sequence in context: A042887 A053908 A149165 * A070761 A149166 A149167 Adjacent sequences: A083856 A083857 A083858 * A083860 A083861 A083862 KEYWORD easy,nonn AUTHOR Paul Barry, May 06 2003 STATUS approved

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Last modified June 5 05:23 EDT 2023. Contains 363130 sequences. (Running on oeis4.)