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

Author

Wolfdieter Lang, Nov 29 2002

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A078363.

Programs

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

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

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 *)

PARI

my(x='x+O('x^20)); Vec(1/(1-13*x+x^2)) \\ G. C. Greubel, May 25 2019

Sage

[lucas_number1(n,13,1) for n in range(1,20)] # Zerinvary Lajos, Jun 25 2008

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).