A083859 Main diagonal of generalized Fibonacci array A083856.

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

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>>^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