A057087 Scaled Chebyshev U-polynomials evaluated at i. Generalized Fibonacci sequence.

1, 4, 20, 96, 464, 2240, 10816, 52224, 252160, 1217536, 5878784, 28385280, 137056256, 661766144, 3195289600, 15428222976, 74494050304, 359689093120, 1736732573696, 8385686667264, 40489676963840, 195501454524416

Offset

0,2

Comments

a(n) gives the length of the word obtained after n steps with the substitution rule 0->1111, 1->11110, starting from 0. The number of 1's and 0's of this word is 4*a(n-1) and 4*a(n-2), respectively.

Inverse binomial transform of odd Pell bisection A001653. With a leading zero, inverse binomial transform of even Pell bisection A001542, divided by 2. - Paul Barry, May 16 2003

For positive n, a(n) equals the permanent of the n X n tridiagonal matrix with 4's along the main diagonal, and 2's along the superdiagonal and the subdiagonal. - John M. Campbell, Jul 19 2011

Pisano period lengths: 1, 1, 8, 1, 3, 8, 6, 1, 24, 3, 120, 8, 21, 6, 24, 1, 16, 24, 360, 3, ... . - R. J. Mathar, Aug 10 2012

Exponential convolution of Pell numbers (A000129) and companion Pell numbers (A002203), divided by 2 and leading zero dropped. - Vladimir Reshetnikov, Oct 07 2016

Links

Indranil Ghosh, Table of n, a(n) for n = 0..1459

Martin Burtscher, Igor Szczyrba, Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.

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=4, q=4.

Tanya Khovanova, Recursive Sequences.

Wolfdieter Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38 (2000) 408-419; Eqs.(39) and (45),rhs, m=4.

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (4,4)

Formula

a(n) = 4*(a(n-1) + a(n-2)), a(-1)=0, a(0)=1.

G.f.: 1/(1 - 4*x - 4*x^2).

a(n) = S(n, 2*i)*(-2*i)^n with S(n, x) := U(n, x/2), Chebyshev's polynomials of the 2nd kind, A049310.

a(n) = Sum_{k=0..n} 3^k*A063967(n,k). - Philippe Deléham, Nov 03 2006

a(n) = A000129(n+1)*A000079(n). - R. J. Mathar, Jul 08 2009

From Johannes W. Meijer, Aug 01 2010: (Start)

Limit_{k->oo} a(n+k)/a(k) = A084128(n) + 2*A057087(n-1)*sqrt(2);

Limit_{n->oo} A084128(n)/A057087(n-1) = sqrt(2). (End)

a(n) = 4^n*hypergeom([1/2-n/2, -n/2], [-n], -1)) for n>=1. - Peter Luschny, Dec 17 2015

Maple

A057087 := n -> `if`(n=0, 1, 4^n*hypergeom([1/2-n/2, -n/2], [-n], -1)):
seq(simplify(A057087(n)), n=0..21); # Peter Luschny, Dec 17 2015

Mathematica

Table[Fibonacci[n + 1, 2] 2^n, {n, 0, 20}] (* Vladimir Reshetnikov, Oct 08 2016 *)
LinearRecurrence[{4, 4}, {1, 4}, 30] (* Harvey P. Dale, Aug 17 2017 *)

Prog

(PARI) a(n)=if(n<0, 0, (2*I)^n*subst(I*poltchebi(n+1)+poltchebi(n), 'x, -I)/2) /* Michael Somos, Sep 16 2005 */
(PARI) Vec(1/(1-4*x-4*x^2) + O(x^100)) \\ Altug Alkan, Dec 17 2015
(Sage) [lucas_number1(n, 4, -4) for n in range(1, 23)] # Zerinvary Lajos, Apr 23 2009
(Magma) I:=[1, 4]; [n le 2 select I[n] else 4*Self(n-1) + 4*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 16 2018

Crossrefs

Pairwise sums are in A086347.

Appears in A086346, A086347 and A086348. - Johannes W. Meijer, Aug 01 2010

Sequence in context: A008353 A250162 A296665 * A151254 A232493 A240778

Adjacent sequences: A057084 A057085 A057086 * A057088 A057089 A057090

Keyword

nonn,easy

Author

Wolfdieter Lang, Aug 11 2000

Status

approved