login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A057087 Scaled Chebyshev U-polynomials evaluated at i. Generalized Fibonacci sequence. 38

%I #79 Dec 18 2023 12:18:08

%S 1,4,20,96,464,2240,10816,52224,252160,1217536,5878784,28385280,

%T 137056256,661766144,3195289600,15428222976,74494050304,359689093120,

%U 1736732573696,8385686667264,40489676963840,195501454524416

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

%C 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.

%C 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

%C 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

%C 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

%C Exponential convolution of Pell numbers (A000129) and companion Pell numbers (A002203), divided by 2 and leading zero dropped. - _Vladimir Reshetnikov_, Oct 07 2016

%H Indranil Ghosh, <a href="/A057087/b057087.txt">Table of n, a(n) for n = 0..1459</a>

%H Martin Burtscher, Igor Szczyrba, Rafał Szczyrba, <a href="http://www.emis.de/journals/JIS/VOL18/Szczyrba/sz3.html">Analytic Representations of the n-anacci Constants and Generalizations Thereof</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.

%H A. F. Horadam, <a href="http://www.fq.math.ca/Scanned/5-5/horadam.pdf">Special properties of the sequence W_n(a,b; p,q)</a>, Fib. Quart., 5.5 (1967), 424-434. Case n->n+1, a=0,b=1; p=4, q=4.

%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>.

%H Wolfdieter Lang, <a href="http://www.fq.math.ca/Scanned/38-5/lang.pdf">On polynomials related to powers of the generating function of Catalan's numbers</a>, Fib. Quart. 38 (2000) 408-419; Eqs.(39) and (45),rhs, m=4.

%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,4)

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

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

%F 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.

%F a(n) = Sum_{k=0..n} 3^k*A063967(n,k). - _Philippe Deléham_, Nov 03 2006

%F a(n) = A000129(n+1)*A000079(n). - _R. J. Mathar_, Jul 08 2009

%F From _Johannes W. Meijer_, Aug 01 2010: (Start)

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

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

%F a(n) = 4^n*hypergeom([1/2-n/2, -n/2], [-n], -1)) for n>=1. - _Peter Luschny_, Dec 17 2015

%p A057087 := n -> `if`(n=0, 1, 4^n*hypergeom([1/2-n/2, -n/2], [-n], -1)):

%p seq(simplify(A057087(n)), n=0..21); # _Peter Luschny_, Dec 17 2015

%t Table[Fibonacci[n + 1, 2] 2^n, {n, 0, 20}] (* _Vladimir Reshetnikov_, Oct 08 2016 *)

%t LinearRecurrence[{4,4},{1,4},30] (* _Harvey P. Dale_, Aug 17 2017 *)

%o (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 */

%o (PARI) Vec(1/(1-4*x-4*x^2) + O(x^100)) \\ _Altug Alkan_, Dec 17 2015

%o (Sage) [lucas_number1(n,4,-4) for n in range(1, 23)] # _Zerinvary Lajos_, Apr 23 2009

%o (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

%Y Pairwise sums are in A086347.

%Y Appears in A086346, A086347 and A086348. - _Johannes W. Meijer_, Aug 01 2010

%K nonn,easy

%O 0,2

%A _Wolfdieter Lang_, Aug 11 2000

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 29 02:23 EDT 2024. Contains 371264 sequences. (Running on oeis4.)