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
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 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).
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)
MATHEMATICA
LinearRecurrence[{12, -1}, {1, 13}, 30] (* Harvey P. Dale, Apr 03 2013 *)
PROG
(SageMath) [(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
KEYWORD
nonn,easy,changed
AUTHOR
Wolfdieter Lang, Nov 29 2002
STATUS
approved