A078362 A Chebyshev S-sequence with Diophantine property.

1, 13, 168, 2171, 28055, 362544, 4685017, 60542677, 782369784, 10110264515, 130651068911, 1688353631328, 21817946138353, 281944946167261, 3643466354036040, 47083117656301259, 608437063177880327

Offset

0,2

Comments

a(n) gives the general (positive integer) solution of the Pell equation b^2 - 165*a^2 = +4 with companion sequence b(n)=A078363(n+1), n >= 0.

This is the m=15 member of the m-family of sequences S(n,m-2) = S(2*n+1,sqrt(m))/sqrt(m). The m=4..14 (nonnegative) sequences are: A000027, A001906, A001353, A004254, A001109, A004187, A001090, A018913, A004189, A004190 and A004191. The m=1..3 (signed) sequences are A049347, A056594, A010892.

For positive n, a(n) equals the permanent of the tridiagonal matrix of order n with 13's along the main diagonal, and i's along the superdiagonal and the subdiagonal (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,...,12}. - Milan Janjic, Jan 23 2015

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..900

A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424-434. Case a=0,b=1; p=13, q=-1.

M. Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.

Tanya Khovanova, Recursive Sequences

W. Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38 (2000) 408-419. Eq.(44), lhs, m=15.

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (13,-1).

Formula

a(n) = 13*a(n-1) - a(n-2), n >= 1; a(-1)=0, a(0)=1.

a(n) = S(2*n+1, sqrt(15))/sqrt(15) = S(n, 13), where S(n, x) = U(n, x/2), Chebyshev polynomials of the 2nd kind, A049310.

a(n) = (ap^(n+1) - am^(n+1))/(ap-am) with ap = (13+sqrt(165))/2 and am = (13-sqrt(165))/2.

G.f.: 1/(1 - 13*x + x^2).

a(n) = Sum_{k=0..n} A101950(n,k)*12^k. - Philippe Deléham, Feb 10 2012

Product {n >= 0} (1 + 1/a(n)) = (1/11)*(11 + sqrt(165)). - Peter Bala, Dec 23 2012

Product {n >= 1} (1 - 1/a(n)) = (1/26)*(11 + sqrt(165)). - Peter Bala, Dec 23 2012

For n >= 1, a(n) = U(n-1,13/2), where U(k,x) represents Chebyshev polynomial of the second order.

a(n) = sqrt((A078363(n+1)^2 - 4)/165), n>=0, (Pell equation d=165, +4).

Mathematica

CoefficientList[Series[1/(1 - 13 x + x^2), {x, 0, 20}], x] (* Vincenzo Librandi, Dec 24 2012 *)
LinearRecurrence[{13, -1}, {1, 13}, 20] (* Harvey P. Dale, Feb 07 2019 *)

Prog

(Sage) [lucas_number1(n, 13, 1) for n in range(1, 20)] # Zerinvary Lajos, Jun 25 2008
(Magma) I:=[1, 13, 168]; [n le 3 select I[n] else 13*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Dec 24 2012
(PARI) my(x='x+O('x^20)); Vec(1/(1-13*x+x^2)) \\ G. C. Greubel, May 25 2019
(GAP) a:=[1, 13, 168];; for n in [4..20] do a[n]:=13*a[n-1]-a[n-2]; od; a; # G. C. Greubel, May 25 2019

Crossrefs

Cf. A078363.

Sequence in context: A277412 A368657 A171318 * A157381 A084970 A209226

Adjacent sequences: A078359 A078360 A078361 * A078363 A078364 A078365

Keyword

nonn,easy

Author

Wolfdieter Lang, Nov 29 2002

Status

approved