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
Indranil Ghosh, Table of n, a(n) for n = 0..723
Tanya Khovanova, Recursive Sequences
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_{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
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 *)
PROG
(Sage) [lucas_number1(n, 24, 1) for n in range(1, 20)] # Zerinvary Lajos, Jun 25 2008
(PARI) vector(21, n, polchebyshev(n-1, 2, 12) ) \\ G. C. Greubel, Dec 22 2019
(Magma) R<x>:=PowerSeriesRing(Integers(), 20); Coefficients(R!( 1/(1-24*x+x^2) )); // G. C. Greubel, Dec 22 2019
(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
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.
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Nov 29 2002
STATUS
approved