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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 nXn 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

LINKS

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

Index entries for sequences related to linear recurrences with constant coefficients

Tanya Khovanova, Recursive Sequences

Index entries for sequences related to Chebyshev polynomials.

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 September 16 03:31 EDT 2014. Contains 246794 sequences.