A106515 A Fibonacci-Pell convolution.

1, 2, 6, 15, 38, 94, 231, 564, 1372, 3329, 8064, 19512, 47177, 114010, 275430, 665247, 1606534, 3879302, 9366735, 22615356, 54601628, 131825377, 318263328, 768369744, 1855031473, 4478479058, 10812064614, 26102729679, 63017720390

Offset

0,2

Comments

Diagonal sums of A106513.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (3,0,-3,-1).

Formula

G.f.: (1-x)/((1-x-x^2)*(1-2*x-x^2)).

a(n) = Sum_{k=0..n} Fibonacci(n-k-1)*Pell(k+1).

a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..floor((n-k+1)/2)} binomial(n-k+1, 2*j+k+1)*2^j.

a(n) = Pell(n) + Pell(n+1) - Fibonacci(n). - Ralf Stephan, Jun 02 2007

a(n) = 3*a(n-1) - 3*a(n-3) - a(n-4). - Wesley Ivan Hurt, May 27 2021

Mathematica

Table[Fibonacci[n, 2] + Fibonacci[n+1, 2] - Fibonacci[n], {n, 0, 30}] (* Vladimir Reshetnikov, Sep 27 2016 *)

Prog

(Magma)
Pell:= func< n | Round(((1+Sqrt(2))^n - (1-Sqrt(2))^n)/(2*Sqrt(2))) >;
[Pell(n) + Pell(n+1) - Fibonacci(n): n in [0..30]]; // G. C. Greubel, Aug 05 2021
(Sage) [lucas_number1(n+1, 2, -1) + lucas_number1(n, 2, -1) - lucas_number1(n, 1, -1) for n in (0..30)] # G. C. Greubel, Aug 05 2021

Crossrefs

Cf. A000045, A000129, A106513.

Sequence in context: A034518 A260787 A290762 * A153122 A109545 A191634

Adjacent sequences: A106512 A106513 A106514 * A106516 A106517 A106518

Keyword

easy,nonn

Author

Paul Barry, May 05 2005

Status

approved