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

Author

Paul Barry, May 05 2005

Keywords

• easy
• nonn

Comments

Diagonal sums of A106513.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Tamás Szakács, Linear recursive sequences and factorials, Ph. D. Thesis, Univ. Debrecen (Hungary, 2024). See pp. 28, 51, 58.
• Index entries for linear recurrences with constant coefficients, signature (3,0,-3,-1).

Crossrefs

Cf. A000045, A000129, A106513.

Programs

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

Mathematica

Table[Fibonacci[n, 2] + Fibonacci[n+1, 2] - Fibonacci[n], {n, 0, 30}] (* Vladimir Reshetnikov, Sep 27 2016 *)
LinearRecurrence[{3,0,-3,-1},{1,2,6,15},30] (* Harvey P. Dale, Feb 08 2025 *)

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

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