A122558 a(0)=1, a(1)=3, a(n) = 4*a(n-1) + 3*a(n-2) for n > 1.

1, 3, 15, 69, 321, 1491, 6927, 32181, 149505, 694563, 3226767, 14990757, 69643329, 323545587, 1503112335, 6983086101, 32441681409, 150715983939, 700188979983, 3252903871749, 15112182426945, 70207441323027, 326166312572943

Offset: 0

Author

Philippe Deléham, Sep 20 2006, Sep 22 2006

Keywords

• nonn

Comments

a(n) is the number of compositions of n when there are 3 types of 1 and 6 types of other natural numbers. - Milan Janjic, Aug 13 2010

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,3).

Crossrefs

Cf. A122542.

Programs

Magma

[n le 2 select 3^(n-1) else 4*Self(n-1) +3*Self(n-2): n in [1..41]]; // G. C. Greubel, Oct 27 2024

Mathematica

LinearRecurrence[{4,3},{1,3},30] (* Harvey P. Dale, Mar 18 2023 *)

PARI

Vec((1-x)/(1-4*x-3*x^2) + O(x^30)) \\ Michel Marcus, Feb 04 2022

SageMath

A122558= BinaryRecurrenceSequence(4,3,1,3)
[A122558(n) for n in range(41)] # G. C. Greubel, Oct 27 2024

Formula

G.f.: (1-x)/(1-4*x-3*x^2).

a(n) = Sum_{k=0..n} 3^k*A122542(n,k).

Limit_{n->infinity} a(n+1)/a(n) = 2 + sqrt(7) = 4.645751311064....

a(n) = ((7+sqrt(7))/14)*(2+sqrt(7))^n + ((7-sqrt(7))/14)*(2-sqrt(7))^n. - Richard Choulet, Nov 20 2008

Extensions

Corrected by T. D. Noe, Nov 07 2006