OFFSET
0,5
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).
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
MATHEMATICA
LinearRecurrence[{1, 1, 0, 1}, {1, 1, 1, 1}, 51] (* G. C. Greubel, Oct 23 2024 *)
PROG
(PARI) Vec((1 - x^2 - x^3) / ((1 + x)*(1 - 2*x + x^2 - x^3)) + O(x^40)) \\ Colin Barker, Apr 25 2020
(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
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Joseph Damico, Apr 21 2019
STATUS
approved