A103425 a(n) = 3*a(n-1) + a(n-2) - 3*a(n-3).

1, 3, 5, 15, 41, 123, 365, 1095, 3281, 9843, 29525, 88575, 265721, 797163, 2391485, 7174455, 21523361, 64570083, 193710245, 581130735, 1743392201, 5230176603, 15690529805, 47071589415, 141214768241, 423644304723, 1270932914165

Offset: 0

Author

Paul Barry, Feb 05 2005

Keywords

• easy
• nonn

Comments

Binomial transform of A103424.

This is a (3, 1, -3) weighted tribonacci sequence, cf. A102001. The current sequence contains primes, including 3, 5, 41, 21523361. Is there an (a, b, c) weighted tribonacci sequence with a, b, c relatively prime which is prime-free? The general linear third-order recurrence equation x(n) = a*x(n-1) + b*x(n-2) + c*x(n-3) has a solution in terms of roots of a cubic polynomial, see Weisstein. - Jonathan Vos Post, Feb 05 2005

Links

• Eric Weisstein's World of Mathematics, Linear Recurrence Equation.
• Index entries for linear recurrences with constant coefficients, signature (3,1,-3).

Formula

G.f.: (1-5x^2)/((1-x^2)(1-3x)).

E.g.f.: exp(x)(1+sinh(2x)).

a(n) = 1 + (3^n - (-1)^n)/2.