A377728 Convolution of Leonardo numbers with Jacobsthal numbers.

0, 1, 2, 7, 16, 39, 86, 189, 402, 847, 1760, 3631, 7438, 15165, 30794, 62343, 125904, 253783, 510758, 1026685, 2061730, 4136991, 8295872, 16627167, 33311646, 66716029, 133582106, 267406999, 535206832, 1071049287, 2143127030, 4287918141, 8578528818, 17161414255

Offset: 0

Author

Prabha Sivaramannair, Nov 05 2024

Keywords

• nonn
• easy

Links

• Prabha Sivaraman Nair, Convolution Identities of p-numbers, Math. Montis. (2024), pp. 26-43.
• Index entries for linear recurrences with constant coefficients, signature (3,0,-5,1,2).

Crossrefs

Cf. A001045, A001595.

Programs

Mathematica

LinearRecurrence[{3, 0, -5, 1, 2}, {0, 1, 2, 7, 16}, 34] (* Amiram Eldar, Nov 07 2024 *)

Python

from sympy import fibonacci
def A377728(n): return 1-(fibonacci(n+2)<<2)+(m:=(4<>1 # Chai Wah Wu, Nov 09 2024

Formula

a(n) = Sum_{i=0..n} L(i)*J(n-i) where L = A001595 and J = A001045.

a(n) = (3*J(n+2) - 2*L(n+1) - 1)/2 where L = A001595 and J = A001045.

G.f.: -x*(x^2-x+1)/((x-1)*(2*x-1)*(x+1)*(x^2+x-1)). - Alois P. Heinz, Nov 05 2024

E.g.f.: 2*cosh(2*x) + sinh(x) + 2*sinh(2*x) - 2*exp(x/2)*(5*cosh(sqrt(5)*x/2) + 3*sqrt(5)*sinh(sqrt(5)*x/2))/5. - Stefano Spezia, Nov 06 2024