A099255 Expansion of g.f. (7+6*x-6*x^2-3*x^3)/((x^2+x-1)*(x^2-x-1)).

7, 6, 15, 15, 38, 39, 99, 102, 259, 267, 678, 699, 1775, 1830, 4647, 4791, 12166, 12543, 31851, 32838, 83387, 85971, 218310, 225075, 571543, 589254, 1496319, 1542687, 3917414, 4038807, 10255923, 10573734, 26850355, 27682395, 70295142, 72473451

Offset: 0

Author

Creighton Dement, Oct 09 2004

Keywords

• nonn
• easy

Comments

One of two sequences involving the Lucas/Fibonacci numbers.

This sequence consists of pairs of numbers more or less close to each other with "jumps" in between pairs. "pos((Ex)^n)" sums up over all floretion basis vectors with positive coefficients for each n. The following relations appear to hold: a(2n) - (a(2n-1) + a(2n-2)) = 2*Luc(2n) a(2n+1) - a(2n) = Fib(2n), apart from initial term a(2n+1)/a(2n-1) -> 2 + golden ratio phi a(2n)/a(2n-2) -> 2 + golden ratio phi An identity: (1/2)a(n) - (1/2)A099256(n) = ((-1)^n)A000032(n)

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (0,3,0,-1).

Crossrefs

Cf. A099256, A000032.

Programs

Mathematica

LinearRecurrence[{0,3,0,-1},{7,6,15,15},40] (* Harvey P. Dale, Dec 29 2012 *)

Formula

a(n) = 2*pos((Ex)^n)

a(0) = 7, a(1) = 6, a(2) = a(3) = 15, a(n+4) = 3a(n+2) - a(n).

a(2n) = A022097(2n+1), a(2n+1) = A022086(2n+3).

a(n) = A061084(n+1)+A013655(n+2). [R. J. Mathar, Nov 30 2008]

Extensions

More terms from Creighton Dement, Apr 19 2005