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A077423 Chebyshev sequence U(n,12)=S(n,24) with Diophantine property. 1
1, 24, 575, 13776, 330049, 7907400, 189447551, 4538833824, 108742564225, 2605282707576, 62418042417599, 1495427735314800, 35827847605137601, 858372914787987624, 20565122107306565375, 492704557660569581376 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

b(n)^2 - 143*a(n)^2 = 1 with the companion sequence b(n)=A077424(n+1).

For positive n, a(n) equals the permanent of the n X n tridiagonal matrix with 24's along the main diagonal, and i's along the subdiagonal and the superdiagonal (i is the imaginary unit). - John M. Campbell, Jul 08 2011

For n>=1, a(n) equals the number of 01-avoiding words of length n-1 on alphabet {0,1,...,23}. Milan Janjic, Jan 25 2015

LINKS

Table of n, a(n) for n=0..15.

Tanya Khovanova, Recursive Sequences

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (24,-1).

FORMULA

a(n)=24*a(n-1) - a(n-2), a(-1) := 0, a(0)=1.

a(n)= S(n, 24) with S(n, x) := U(n, x/2) Chebyshev's polynomials of the 2nd kind. See A049310.

a(n)= (ap^(n+1) - am^(n+1))/(ap - am) with ap := 12+sqrt(143) and am := 12-sqrt(143).

a(n)= sum(((-1)^k)*binomial(n-k, k)*24^(n-2*k), k=0..floor(n/2)).

a(n)=sqrt((A077424(n+1)^2 - 1)/143).

G.f.: 1/(1-24*x+x^2). - Philippe Deléham, Nov 18 2008

a(n) = Sum_{k, 0<=k<=n} A101950(n,k)*23^k. - Philippe Deléham, Feb 10 2012

Product {n >= 0} (1 + 1/a(n)) = 1/11*(11 + sqrt(143)). - Peter Bala, Dec 23 2012

Product {n >= 1} (1 - 1/a(n)) = 1/24*(11 + sqrt(143)). - Peter Bala, Dec 23 2012

MATHEMATICA

lst={}; Do[AppendTo[lst, GegenbauerC[n, 1, 12]], {n, 0, 8^2}]; lst (* Vladimir Joseph Stephan Orlovsky, Sep 11 2008 *)

PROG

(Sage)[lucas_number1(n, 24, 1) for n in xrange(1, 20)] # Zerinvary Lajos, Jun 25 2008

CROSSREFS

Sequence in context: A007109 A158538 A171329 * A059061 A206991 A206933

Adjacent sequences:  A077420 A077421 A077422 * A077424 A077425 A077426

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Nov 29 2002

STATUS

approved

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Last modified May 30 04:53 EDT 2016. Contains 273513 sequences.