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 A057087 Scaled Chebyshev U-polynomials evaluated at i. Generalized Fibonacci sequence. 38
 1, 4, 20, 96, 464, 2240, 10816, 52224, 252160, 1217536, 5878784, 28385280, 137056256, 661766144, 3195289600, 15428222976, 74494050304, 359689093120, 1736732573696, 8385686667264, 40489676963840, 195501454524416 (list; graph; refs; listen; history; text; internal format)
 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. W. 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 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) = -(1/8)*sqrt(2)*(2 - 2*sqrt(2))^(n+1)+(1/8)*(2 + 2*sqrt(2))^(n+1)*sqrt(2). - Paolo P. Lava, Nov 20 2008 a(n) = A000129(n+1)*A000079(n). - R. J. Mathar, Jul 08 2009 From Johannes W. Meijer, Aug 01 2010: (Start) Limit(a(n+k)/a(k), k=infinity) = A084128(n) + 2*A057087(n-1)*sqrt(2); Limit(A084128(n)/A057087(n-1), n=infinity) = 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

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Last modified September 28 01:47 EDT 2022. Contains 357063 sequences. (Running on oeis4.)