A306276 a(0) = a(1) = a(2) = a(3) = 1; thereafter a(n) = a(n-2) + a(n-3) + a(n-4).

1, 1, 1, 1, 3, 3, 5, 7, 11, 15, 23, 33, 49, 71, 105, 153, 225, 329, 483, 707, 1037, 1519, 2227, 3263, 4783, 7009, 10273, 15055, 22065, 32337, 47393, 69457, 101795, 149187, 218645, 320439, 469627, 688271, 1008711, 1478337, 2166609, 3175319, 4653657, 6820265

Offset: 0

Author

Joseph Damico, Feb 02 2019

Keywords

• nonn

Comments

The characteristic equation of this sequence is x^4 = x^2 + x + 1. The characteristic equation of A000930 is x^3 = x^2 + 1 [1], which can be rewritten as x^4 = x^3 + x [2]. By substituting the value of x^3 from equation [1] in equation [2], we get x^4 = (x^2 + 1) + x, which is the characteristic equation for this sequence. Hence the ratio a(n+1)/a(n) has the same limit as the A000930 sequence does, about 1.465571231.

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..6025
• Anthony Shannon, François Dubeau, Mine Uysal, and Engin Özkan, A Difference Equation Model of Infectious Disease, Int. J. Bioautomation (2022) Vol. 26, No. 4, 339-352.

Crossrefs

Cf. A000930, A092526.

Programs

Mathematica

Nest[Append[#, Total@ #[[-4 ;; -2]] ] &, {1, 1, 1, 1}, 40] (* or *)
CoefficientList[Series[(x^3 - x - 1)/(x^4 + x^3 + x^2 - 1), {x, 0, 43}], x] (* Michael De Vlieger, Feb 09 2019 *)

Formula

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

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

Lim_{n->infinity} a(n+1)/a(n) = A092526.