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A029548 Expansion of 1/(1-32*x+x^2). 4
1, 32, 1023, 32704, 1045505, 33423456, 1068505087, 34158739328, 1092011153409, 34910198169760, 1116034330278911, 35678188370755392, 1140585993533893633, 36463073604713840864, 1165677769357309014015 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

From Bruno Berselli, Nov 21 2011: (Start)

A Diophantine property of these numbers: ((a(n+1)-a(n-1))/2)^2 - 255*a(n)^2 = 1.

More generally, for t(m)=m+sqrt(m^2-1) and u(n)=(t(m)^(n+1)-1/t(m)^(n+1))/(t(m)-1/t(m)), we can verify that ((u(n+1)-u(n-1))/2)^2-(m^2-1)*u(n)^2=1. (End)

a(n) equals the number of 01-avoiding words of length n on alphabet {0,1,...,31}. - Milan Janjic, Jan 26 2015

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..600

Tanya Khovanova, Recursive Sequences

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (32,-1).

FORMULA

a(n) = 32*a(n-1) - a(n-2), a(-1)=0, a(0)=1.

a(n) = S(n, 32) with S(n, x) = U(n, x/2) Chebyshev's polynomials of the 2nd kind. See A049310. - Wolfdieter Lang, Nov 29 2002

a(n) = (ap^(n+1) - am^(n+1))/(ap - am) with ap=16+sqrt(255) and am=16-sqrt(255).

a(n) = Sum_{k=0..floor(n/2)} (-1)^k*binomial(n-k, k)*32^(n-2*k).

a(n) = Sum_{k, 0<=k<=n} A101950(n,k)*31^k. - Philippe Deléham, Feb 10 2012

Product {n >= 0} (1 + 1/a(n)) = 1/15*(15 + sqrt(255)). - Peter Bala, Dec 23 2012

Product {n >= 1} (1 - 1/a(n)) = 1/32*(15 + sqrt(255)). - Peter Bala, Dec 23 2012

MATHEMATICA

lst={}; Do[AppendTo[lst, GegenbauerC[n, 1, 16]], {n, 0, 8^2}]; lst (* Vladimir Joseph Stephan Orlovsky, Sep 11 2008 *)

CoefficientList[Series[1/(1 - 32 x + x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 24 2012 *)

PROG

(Sage) [lucas_number1(n, 32, 1) for n in xrange(1, 16)] # Zerinvary Lajos, Nov 07 2009

(MAGMA) I:=[1, 32, 1023]; [n le 3 select I[n] else 32*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Dec 24 2012

CROSSREFS

Cf. A200441, A200442.

Sequence in context: A065552 A158617 A171337 * A016745 A189267 A223672

Adjacent sequences:  A029545 A029546 A029547 * A029549 A029550 A029551

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified August 18 11:04 EDT 2017. Contains 290714 sequences.