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

0,2

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

Links

Ivan Panchenko, Table of n, a(n) for n = 0..200

Andersen, K., Carbone, L. and Penta, D., 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

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (12,-1).

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).

Mathematica

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

Prog

(Sage) [(lucas_number2(n, 12, 1)-lucas_number2(n-1, 12, 1))/10 for n in range(1, 18)] # Zerinvary Lajos, Nov 10 2009
(PARI) x='x+O('x^30); Vec((1+x)/(1-12*x+x^2)) \\ G. C. Greubel, Jan 18 2018
(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

Crossrefs

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

Sequence in context: A108366 A204766 A163415 * A192092 A297454 A102146

Adjacent sequences: A077413 A077414 A077415 * A077417 A077418 A077419

Keyword

nonn,easy

Author

Wolfdieter Lang, Nov 29 2002

Status

approved