A006684 Convolve Fibonacci and Pell numbers.

0, 0, 1, 3, 9, 24, 62, 156, 387, 951, 2323, 5652, 13716, 33228, 80405, 194415, 469845, 1135092, 2741626, 6620928, 15987663, 38603019, 93204647, 225030024, 543293352, 1311663096, 3166694569, 7645173627, 18457238241, 44559967920, 107577688310, 259716176580

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Define a triangle T(r,c) by T(n,0) = T(n,n) = A000045(n) and T(r,c) = T(r-1,c) + T(r-1,c-1) + T(r-2,c-1). The sum of the terms in the first n rows is 2*a(n+1). - J. M. Bergot, Apr 07 2013

Links

• T. D. Noe, Table of n, a(n) for n = 0..500
• A. Özkoç, Some algebraic identities on quadra Fibona-Pell integer sequence, Advances in Difference Equations, 2015:148 (2015).
• Index entries for linear recurrences with constant coefficients, signature (3,0,-3,-1).

Crossrefs

Cf. A000045, A000129, A106515 (first differences).

Programs

Magma

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

Maple

with(combinat):seq(fibonacci(i,2)-fibonacci(i, 1),i=0..27); # Zerinvary Lajos, Mar 20 2008

Mathematica

LinearRecurrence[{3,0,-3,-1}, {0,0,1,3}, 50] (* T. D. Noe, Apr 16 2013 *)
Table[Fibonacci[n, 2] - Fibonacci[n], {n, 0, 30}] (* Vladimir Reshetnikov, Sep 27 2016 *)

Sage

[lucas_number1(n, 2, -1) - lucas_number1(n, 1, -1) for n in (0..30)] # G. C. Greubel, Aug 05 2021

Formula

a(n) = Pell(n) - Fibonacci(n).

G.f.: x^2/( (1-x-x^2)*(1-2*x-x^2) ). - Joerg Arndt, Apr 17 2013

a(n) = 3*a(n-1) - 3*a(n-3) - a(n-4) with a(0) = a(1) = 0, a(2) = 1, a(3) = 3. - Taras Goy, Mar 12 2019