A077423 Chebyshev sequence U(n,12)=S(n,24) with Diophantine property.

1, 24, 575, 13776, 330049, 7907400, 189447551, 4538833824, 108742564225, 2605282707576, 62418042417599, 1495427735314800, 35827847605137601, 858372914787987624, 20565122107306565375, 492704557660569581376

Offset: 0

Author

Wolfdieter Lang, Nov 29 2002

Keywords

• nonn
• easy

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

• Indranil Ghosh, Table of n, a(n) for n = 0..723
• Tanya Khovanova, Recursive Sequences
• Index entries for sequences related to Chebyshev polynomials.
• Index entries for linear recurrences with constant coefficients, signature (24,-1).

Crossrefs

Chebyshev sequence U(n, m): A000027 (m=1), A001353 (m=2), A001109 (m=3), A001090 (m=4), A004189 (m=5), A004191 (m=6), A007655 (m=7), A077912 (m=8), A049660 (m=9), A075843 (m=10), A077421 (m=11), this sequence (m=12), A097309 (m=13), A097311 (m=14), A097313 (m=15), A029548 (m=16), A029547 (m=17), A144128 (m=18), A078987 (m=19), A097316 (m=33).

Cf. A323182.

Programs

GAP

a:=[1,24];; for n in [3..20] do a[n]:=24*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Dec 22 2019

Magma

R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( 1/(1-24*x+x^2) )); // G. C. Greubel, Dec 22 2019

Maple

seq( simplify(ChebyshevU(n, 12)), n=0..20); # G. C. Greubel, Dec 22 2019

Mathematica

Table[GegenbauerC[n, 1, 12], {n,0,20}] (* Vladimir Joseph Stephan Orlovsky, Sep 11 2008 *)
ChebyshevU[Range[21] -1, 12] (* G. C. Greubel, Dec 22 2019 *)

PARI

vector(21, n, polchebyshev(n-1, 2, 12) ) \\ G. C. Greubel, Dec 22 2019

Sage

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

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_{k=0..floor(n/2)} (-1)^k*binomial(n-k, k)*24^(n-2*k).

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..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