A077416 Chebyshev S-sequence with Diophantine property.

1, 13, 155, 1847, 22009, 262261, 3125123, 37239215, 443745457, 5287706269, 63008729771, 750817050983, 8946795882025, 106610733533317, 1270382006517779, 15137973344680031, 180385298129642593

Offset: 0

Author

Wolfdieter Lang, Nov 29 2002

Keywords

• nonn
• easy

Comments

7*b(n)^2 - 5*a(n)^2 = 2 with companion sequence b(n) = A077417(n), n>=0.

a(n) = L(n,-12)*(-1)^n, where L is defined as in A108299; see also A077417 for L(n,+12). - Reinhard Zumkeller, Jun 01 2005

The aerated sequence (b(n))n>=1 = [1, 0, 13, 0, 155, 0, 1857, 0, ...] is a fourth-order linear divisibility sequence; that is, if n | m then b(n) | b(m). It is the case P1 = 0, P2 = -10, Q = -1 of the 3-parameter family of divisibility sequences found by Williams and Guy. See A100047. - Peter Bala, May 12 2025

Links

• Ivan Panchenko, Table of n, a(n) for n = 0..200
• K. Andersen, L. Carbone, and D. Penta, Kac-Moody Fibonacci sequences, hyperbolic golden ratios, and real quadratic fields, Journal of Number Theory and Combinatorics, Vol 2, No. 3 pp 245-278, 2011. See Section 9.
• Alex Fink, Richard K. Guy, and Mark Krusemeyer, Partitions with parts occurring at most thrice, Contributions to Discrete Mathematics, Vol 3, No 2 (2008), pp. 76-114. See Section 13.
• Tanya Khovanova, Recursive Sequences
• H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
• H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences Integers, Volume 12A (2012) The John Selfridge Memorial Volume.
• Index entries for sequences related to Chebyshev polynomials.
• Index entries for linear recurrences with constant coefficients, signature (12,-1).

Crossrefs

Cf. A054320(n-1) with companion A072256(n), n>=1.

Programs

Magma

I:=[1, 13]; [n le 2 select I[n] else 12*Self(n-1) - Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 18 2018

Mathematica

LinearRecurrence[{12,-1},{1,13},30] (* Harvey P. Dale, Apr 03 2013 *)

PARI

x='x+O('x^30); Vec((1+x)/(1-12*x+x^2)) \\ G. C. Greubel, Jan 18 2018

Sage

[(lucas_number2(n,12,1)-lucas_number2(n-1,12,1))/10 for n in range(1, 18)] # Zerinvary Lajos, Nov 10 2009

Formula

a(n) = 12*a(n-1) - a(n-2), a(-1)=-1, a(0)=1.

a(n) = S(n, 12) + S(n-1, 12) = S(2*n, sqrt(14)) with S(n, x) := U(n, x/2) Chebyshev's polynomials of the second kind. See A049310. S(-1, x)=0, S(n, 12) = A004191(n).

G.f.: (1+x)/(1-12*x+x^2).

a(n) = (ap^(2*n+1) - am^(2*n+1))/(ap - am) with ap := (sqrt(7)+sqrt(5))/sqrt(2) and am := (sqrt(7)-sqrt(5))/sqrt(2).

a(n) = Sum_{k=0..n} (-1)^k * binomial(2*n-k,k) * 14^(n-k).

a(n) = sqrt((7*A077417(n)^2 - 2)/5).

From Peter Bala, May 09 2025: (Start)

a(n) = Dir(n, 6), where Dir(n, x) denotes the n-th row polynomial of the triangle A244419.

a(n)^2 - 12*a(n)*a(n+1) + a(n+1)^2 = 14.

More generally, for real x, a(n+x)^2 - 12*a(n+x)*a(n+x+1) + a(n+x+1)^2 = 14, where a(n) := (ap^(2*n+1) - am^(2*n+1))/(ap - am), ap := sqrt(7/2) + sqrt(5/2) and am := sqrt(7/2) - sqrt(5/2), as given above.

Sum_{n >= 1} (-1)^(n+1)/(a(n) - 1/a(n)) = 1/14 (telescoping series).

Product_{n >= 1} (a(n) + 1)/(a(n) - 1) = sqrt(7/5) (telescoping product). (End)