cp's OEIS Frontend

A083859 Main diagonal of generalized Fibonacci array A083856.

%I A083859 #38 Sep 08 2022 08:45:10
%S A083859 0,1,1,4,9,41,133,673,2737,15130,72181,430739,2320825,14815529,
%T A083859 88005541,596681296,3843559137,27515587661,189933449365,1428716457761,
%U A083859 10474213334761,82448447397646,637534807917701,5233087759204967,42445677865505425,362213650380301201
%N A083859 Main diagonal of generalized Fibonacci array A083856.
%C A083859 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]
%H A083859 G. C. Greubel, <a href="/A083859/b083859.txt">Table of n, a(n) for n = 0..690</a>
%F A083859 a(n) = (((1 + sqrt(4*n + 1))/2)^n - ((1 - sqrt(4*n + 1))/2)^n)/sqrt(4*n + 1).
%F A083859 a(n) = A193376(n-1,n) for n >= 2. - _R. J. Mathar_, Aug 23 2011
%F A083859 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
%F A083859 a(n) = [x^n] x/(1 - x - n*x^2). - _Ilya Gutkovskiy_, Oct 10 2017
%F A083859 a(n) = Sum_{s = 0..floor((n-1)/2)} binomial(n-1-s, s) * n^s. - _Petros Hadjicostas_, Dec 24 2019
%F A083859 From _G. C. Greubel_, Dec 27 2019: (Start)
%F A083859 a(n) = (sqrt(n))^n * Fibonacci(n, 1/sqrt(n)), with a(0)=0.
%F A083859 a(n) = (-sqrt(n)*i)^(n-1)*ChebyshevU(n-1, i/(2*sqrt(n))), with a(0)=0. (End)
%p A083859 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
%p A083859 # second Maple program:
%p A083859 a:= n-> (<<0|1>, <n|1>>^n)[1, 2]:
%p A083859 seq(a(n), n=0..25);  # _Alois P. Heinz_, Oct 19 2021
%t A083859 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 *)
%t A083859 Table[If[n==0, 0, Round[(Sqrt[n])^(n-1)*Fibonacci[n, 1/Sqrt[n]] ]], {n,0,30}] (* _G. C. Greubel_, Dec 27 2019 *)
%o A083859 (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
%o A083859 (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
%o A083859 (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
%o A083859 (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
%Y A083859 Cf. A083856, A083860, A193376.
%K A083859 easy,nonn
%O A083859 0,4
%A A083859 _Paul Barry_, May 06 2003