A344684 Sum of two consecutive products of Fibonacci and Pell numbers: F(n)*P(n) + F(n+1)*P(n+1).

1, 3, 12, 46, 181, 705, 2757, 10765, 42058, 164280, 641739, 2506789, 9792253, 38251227, 149420064, 583676434, 2280003517, 8906330973, 34790619369, 135901886149, 530870766310

Offset

0,2

Comments

a(n) is the numerator of the continued fraction [1,...,1,2,...,2] with n 1's followed by n 2's.

Links

Table of n, a(n) for n=0..20.

Index entries for linear recurrences with constant coefficients, signature (2,7,2,-1).

Formula

a(n) = F(n)*P(n) + F(n+1)*P(n+1) for F(n) = A000045(n) the Fibonacci numbers and P(n) = A000129(n) the Pell numbers.

a(n) = 2*a(n-1) + 7*a(n-2) + 2*a(n-3) - a(n-4).

G.f.: (1 + x - x^2 - x^3)/(1 - 2*x - 7*x^2 - 2*x^3 + x^4).

a(n) = A001582(n-1) + A001582(n) for n >= 1.

Example

For n=3, a(3)=46 which is F(3)*P(3) + F(4)*P(4) = 2*5 + 3*12 = 46. Also, the continued fraction [1,1,1,2,2,2] with 3 1's followed by 3 2's has numerator 46.

Mathematica

Table[Fibonacci[n] Fibonacci[n, 2] + Fibonacci[n + 1] Fibonacci[n + 1, 2], {n, 0, 30}]

Prog

(PARI) P(n) = ([2, 1; 1, 0]^n)[2, 1]; \\ A000129
a(n) = fibonacci(n)*P(n)+ fibonacci(n+1)*P(n+1); \\ Michel Marcus, Aug 18 2021

Crossrefs

Cf. A000045, A000129, A001582.

Sequence in context: A370478 A108368 A264514 * A216946 A303191 A176310

Adjacent sequences: A344681 A344682 A344683 * A344685 A344686 A344687

Keyword

nonn,easy

Author

Greg Dresden and Hexuan Wang, Aug 17 2021

Status

approved