A030191 Scaled Chebyshev U-polynomial evaluated at sqrt(5)/2.

1, 5, 20, 75, 275, 1000, 3625, 13125, 47500, 171875, 621875, 2250000, 8140625, 29453125, 106562500, 385546875, 1394921875, 5046875000, 18259765625, 66064453125, 239023437500, 864794921875, 3128857421875, 11320312500000, 40957275390625, 148184814453125

Offset

0,2

Comments

Number of (s(0), s(1), ..., s(2n+4)) such that 0 < s(i) < 10 and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n+4, s(0) = 1, s(2n+4) = 5. - Herbert Kociemba, Jun 14 2004

Binomial transform of A002878. - Philippe Deléham, Oct 04 2005

Diagonal of square array A216219. - Philippe Deléham, Mar 15 2013

Lim_{n->oo} a(n+1)/a(n) = 2 + phi = A296184, where phi = A001622. - Wolfdieter Lang, Nov 16 2023~

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Santiago Alzate, Oscar Correa, and Rigoberto Flórez, Fibonacci identities from Jordan Identities, arXiv:2009.02639 [math.NT], 2020.

A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424-434. Case n->n+1, a=0 ,b=1; p=5, q=-5.

Wolfdieter Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38 (2000) 408-419. Eqs. (38) and (45), lhs, m=5.

László Németh and László Szalay, Sequences Involving Square Zig-Zag Shapes, J. Int. Seq., Vol. 24 (2021), Article 21.5.2.

Index entries for linear recurrences with constant coefficients, signature (5,-5).

Index entries for sequences related to Chebyshev polynomials.

Formula

a(n) = (sqrt(5))^n*U(n, sqrt(5)/2).

G.f.: 1/(1 - 5*x + 5*x^2).

a(2*k+1) = 5^(k+1)*Fibonacci(2*k+2).

a(2*k) = 5^k*Lucas(2*k+1).

a(n-1) = Sum_{k=0..n} C(n, k)*Fibonacci(2*k). - Benoit Cloitre, Jun 21 2003

a(n) = 5*a(n-1) - 5*a(n-2). - Benoit Cloitre, Oct 23 2003

a(n-1) = (((5+sqrt(5))/2)^n - ((5-sqrt(5))/2)^n)/sqrt(5) is the 2nd binomial transform of Fibonacci(n), the first binomial transform of Fibonacci(2n) and its n-th term is the n-th term of the third binomial transform of Fibonacci(3n) divided by 2^n. - Paul Barry, Mar 23 2004

a(n) = Sum_{k-0..n} 5^k*A109466(n,k). - Philippe Deléham, Nov 28 2006

a(n) = 5*A039717(n), n>0. - Philippe Deléham, Mar 12 2013

a(n) = A216219(n,n+3) = A216219(n,n+4) = A216219(n+3,n) = A216219(n+4,n). - Philippe Deléham, Mar 15 2013

G.f.: 1/(1-5*x/(1+x/(1-x))). - Philippe Deléham, Mar 15 2013

a(n) = -a(-2-n) * 5^(n+1) for all n in Z. - Michael Somos, Aug 27 2015

E.g.f.: exp((5-sqrt(5))*x/2)*((5 + sqrt(5))*exp(sqrt(5)*x) - 5 + sqrt(5))/(2*sqrt(5)). - Stefano Spezia, Dec 29 2019

a(n) = Sum_{k=0..n} A081567(n-k)*2^k. - Philippe Deléham, Mar 10 2023

Example

G.f. = 1 + 5*x + 20*x^2 + 75*x^3 + 275*x^4 + 1000*x^5 + 3625*x^6 + ...

Maple

seq(coeff(series(1/(1-5*x+5*x^2), x, n+1), x, n), n = 0..30); # G. C. Greubel, Dec 28 2019

Mathematica

Table[MatrixPower[{{2, 1}, {1, 3}}, n][[1]][[2]], {n, 0, 44}] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2010 *)
a[ n_]:= (((5 + Sqrt[5])/2)^(n + 1) - ((5 - Sqrt[5])/2)^(n + 1)) / Sqrt[5] // Expand; (* Michael Somos, Aug 27 2015 *)
Table[If[EvenQ[n], 5^(n/2)*LucasL[n+1], 5^((n+1)/2)*Fibonacci[n+1]], {n, 0, 30}] (* G. C. Greubel, Dec 28 2019 *)

Prog

(Sage) [lucas_number1(n, 5, 5) for n in range(1, 22)] # Zerinvary Lajos, Apr 22 2009
(PARI) {a(n) = imag((quadgen(5) + 2)^(n+1))}; /* Michael Somos, Aug 27 2015 *
(Magma) I:=[1, 5]; [n le 2 select I[n] else 5*(Self(n-1) - Self(n-2)): n in [1..30]]; // G. C. Greubel, Dec 28 2019
(GAP) a:=[1, 5];; for n in [3..30] do a[n]:=5*(a[n-1]-a[n-2]); od; a; # G. C. Greubel, Dec 28 2019

Crossrefs

Cf. A000032, A000045, A001622.

Cf. A002878, A039717, A081567, A109466, A216219.

Sequence in context: A022633 A092490 A094828 * A093131 A224422 A000344

Adjacent sequences: A030188 A030189 A030190 * A030192 A030193 A030194

Keyword

nonn,easy

Author

Wolfdieter Lang

Status

approved