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

%I A057087 #84 Jan 05 2025 19:51:36
%S A057087 1,4,20,96,464,2240,10816,52224,252160,1217536,5878784,28385280,
%T A057087 137056256,661766144,3195289600,15428222976,74494050304,359689093120,
%U A057087 1736732573696,8385686667264,40489676963840,195501454524416
%N A057087 Scaled Chebyshev U-polynomials evaluated at i. Generalized Fibonacci sequence.
%C A057087 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 A057087 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 A057087 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 A057087 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 A057087 Exponential convolution of Pell numbers (A000129) and companion Pell numbers (A002203), divided by 2 and leading zero dropped. - _Vladimir Reshetnikov_, Oct 07 2016
%H A057087 Indranil Ghosh, <a href="/A057087/b057087.txt">Table of n, a(n) for n = 0..1459</a>
%H A057087 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 A057087 A. F. Horadam, <a href="https://web.archive.org/web/2024*/https://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 A057087 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>.
%H A057087 Wolfdieter Lang, <a href="https://web.archive.org/web/2024*/https://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 A057087 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H A057087 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,4)
%F A057087 a(n) = 4*(a(n-1) + a(n-2)), a(-1)=0, a(0)=1.
%F A057087 G.f.: 1/(1 - 4*x - 4*x^2).
%F A057087 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 A057087 a(n) = Sum_{k=0..n} 3^k*A063967(n,k). - _Philippe Deléham_, Nov 03 2006
%F A057087 a(n) = A000129(n+1)*A000079(n). - _R. J. Mathar_, Jul 08 2009
%F A057087 From _Johannes W. Meijer_, Aug 01 2010: (Start)
%F A057087 Limit_{k->oo} a(n+k)/a(k) = A084128(n) + 2*A057087(n-1)*sqrt(2);
%F A057087 Limit_{n->oo} A084128(n)/A057087(n-1) = sqrt(2). (End)
%F A057087 a(n) = 4^n*hypergeom([1/2-n/2, -n/2], [-n], -1) for n>=1. - _Peter Luschny_, Dec 17 2015
%p A057087 A057087 := n -> `if`(n=0, 1, 4^n*hypergeom([1/2-n/2, -n/2], [-n], -1)):
%p A057087 seq(simplify(A057087(n)), n=0..21); # _Peter Luschny_, Dec 17 2015
%t A057087 Table[Fibonacci[n + 1, 2] 2^n, {n, 0, 20}] (* _Vladimir Reshetnikov_, Oct 08 2016 *)
%t A057087 LinearRecurrence[{4,4},{1,4},30] (* _Harvey P. Dale_, Aug 17 2017 *)
%o A057087 (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 A057087 (PARI) Vec(1/(1-4*x-4*x^2) + O(x^100)) \\ _Altug Alkan_, Dec 17 2015
%o A057087 (Sage) [lucas_number1(n,4,-4) for n in range(1, 23)] # _Zerinvary Lajos_, Apr 23 2009
%o A057087 (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 A057087 Pairwise sums are in A086347.
%Y A057087 Appears in A086346, A086347 and A086348. - _Johannes W. Meijer_, Aug 01 2010
%K A057087 nonn,easy
%O A057087 0,2
%A A057087 _Wolfdieter Lang_, Aug 11 2000