A307677 a(0) = a(1) = a(2) = a(3) = 1; thereafter a(n) = a(n-1) + a(n-2) + a(n-4).

1, 1, 1, 1, 3, 5, 9, 15, 27, 47, 83, 145, 255, 447, 785, 1377, 2417, 4241, 7443, 13061, 22921, 40223, 70587, 123871, 217379, 381473, 669439, 1174783, 2061601, 3617857, 6348897, 11141537, 19552035, 34311429, 60212361, 105665327, 185429723, 325406479, 571048563, 1002120369

Offset: 0

Author

Joseph Damico, Apr 21 2019

Keywords

• nonn
• easy

Comments

A079398, A103609, A003269, A306276, A126116, and A000288 are the other six sequences which have characteristic equations of the form x^4 = ax^3 + bx^2 + cx + 1 in which a, b, and c are equal to either 0 or 1 -- but not all three of them are equal to zero. (Each of those sequences begins with 1,1,1,1.)

A005251 has the same characteristic equation, and each successive term is determined by the same operation, namely, a(n) = a(n-1) + a(n-2) + a(n-4). However, it has different starting values: (0,1,1,1) instead of (1,1,1,1).

The characteristic equation of this sequence is x^4 = x^3 + x^2 + 1. Lim_{n->infinity} a(n+1)/a(n) = 1.754877666...

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,1,0,1).

Crossrefs

Cf. A000288, A003269, A005251, A005314, A079398, A103609, A126116, A306276.

Programs

Magma

[n le 4 select 1 else Self(n-1) +Self(n-2) +Self(n-4): n in [1..51]]; // G. C. Greubel, Oct 23 2024

Mathematica

LinearRecurrence[{1,1,0,1}, {1,1,1,1}, 51] (* G. C. Greubel, Oct 23 2024 *)

PARI

Vec((1 - x^2 - x^3) / ((1 + x)*(1 - 2*x + x^2 - x^3)) + O(x^40)) \\ Colin Barker, Apr 25 2020

SageMath

@CachedFunction # a = A307677
def a(n): return 1 if n<4 else a(n-1) +a(n-2) +a(n-3)
[a(n) for n in range(51)] # G. C. Greubel, Oct 23 2024

Formula

From Colin Barker, Apr 25 2020: (Start)

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

a(n) = a(n-1) + a(n-2) + a(n-4) for n>3. (End)

a(n) = (1/5)*((-1)^n + 2*(2*A005314(n+1) - A005314(n) - 2*A005314(n-1))). - G. C. Greubel, Oct 23 2024