A306276 -id:A306276 - cp's OEIS frontend

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

Joseph Damico, Apr 21 2019

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...

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

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

[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

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

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

@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

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

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

SageMath

Formula