O.g.f.: 1/(1 - x - 3*x^2). -
Simon Plouffe in his 1992 dissertation.
a(n) = (( (1 + sqrt(13))/2 )^(n+1) - ( (1 - sqrt(13))/2 )^(n+1))/sqrt(13).
a(0) = 1; a(1) = 1; for n >= 1, a(n+1) = (a(n)^2 - (-3)^n) / a(n-1). -
Philippe Deléham, Mar 07 2004
The i-th term of the sequence is the (1, 2) entry in the i-th power of the 2 X 2 matrix M = ((-1, 1), (1, 2)). -
Simone Severini, Oct 15 2005
a(n) = lower right term in the 2 X 2 matrix [0,3; 1,1]^n. -
Gary W. Adamson, Mar 02 2008
G.f.: G(0)/(2-x), where G(k)= 1 + 1/(1 - x*(13*k - 1)/(x*(13*k + 12) - 2/G(k+1))); (continued fraction). -
Sergei N. Gladkovskii, Jun 18 2013
G.f.: Q(0)/2, where Q(k) = 1 + 1/(1 - x*(4*k+1 + 3*x)/( x*(4*k+3 + 3*x) + 1/Q(k+1) )); (continued fraction). -
Sergei N. Gladkovskii, Sep 08 2013
a(n) = ( Sum_{1 <= k <= n+1, k odd} C(n+1,k)*13^((k-1)/2) )/2^n. -
Vladimir Shevelev, Feb 05 2014
E.g.f.: (1/(a - b))*(a*exp(a*x) - b*exp(b*x)), where 2*a = 1 + sqrt(13) and 2*b = 1 - sqrt(13). -
G. C. Greubel, Aug 30 2015
a(n) = ((i*sqrt(3))^n)*S(n, (-i/sqrt(3))), with the imaginary unit i and the Chebyshev S polynomials (coefficients in
A049310). -
Wolfdieter Lang, Feb 18 2018
a(n) = hypergeom([(1-n)/2, -n/2], [-n], -12) for n >= 1. -
Peter Luschny, Feb 18 2018
a(n) = 3 * (-3)^n * a(-2-n) for all n in Z. -
Michael Somos, Nov 04 2018
a(n) = ( sqrt(3) )^n * Fibonacci(n+1, 1/sqrt(3)). -
G. C. Greubel, Dec 26 2019
a(n) = numerator of the continued fraction 1 + 3/(1 + 3/(1 + 3/ ... + 3/1)) with exactly n 1's, for n>0. -
Greg Dresden and
Alexander Mark, Aug 16 2020
With an initial 0 prepended, the sequence [0, 1, 1, 4, 7, 19, 40, ...] satisfies the congruences a(n*p^k) == e*a(n*p^(k-1)) (mod p^k) for positive integers k and n and all primes p, where e = +1 for the primes p listed in
A296937, e = 0 if p = 13, otherwise e = -1. See Young, Theorem 1, Corollary 1 (i). -
Peter Bala, Dec 28 2022
a(n) = sqrt(-3)^n*S(n, 1/sqrt(-3)) with the S-Chebyshev polynomials (see
A049310), also valid for negative n, using S(-n, x) = -S(n-2, x), for n >= 2, and S(-1, x) = 0. See above the formula in terms of Fibonacci polynomials).
a(n) =
A052533(n+2)/3, for n >= 0. (End)
G.f.: Sum_{n >= 0} x^n * Product_{k = 1..n} (k + 3*x)/(1 + k*x) = Sum_{n >= 0} x^n * Product_{k = 1..n} (1 + 3*k*x)/(1 + 3*k*x^2).
The following products telescope:
Product_{k >= 0} (1 + 3^k/a(2*k+1)) = 1 + sqrt(13).
Product_{k >= 1} (1 - 3^k/a(2*k+1)) = 1/14 * (1 + sqrt(13)).
Product_{k >= 0} (1 + (-3)^k/a(2*k+1)) = (1/13) * (13 + sqrt(13)).
Product_{k >= 1} (1 - (-3)^k/a(2*k+1)) = (1/14) * (13 + sqrt(13)). (End)
