%I #34 Dec 28 2022 01:56:14
%S 1,1,1,1,3,3,5,7,11,15,23,33,49,71,105,153,225,329,483,707,1037,1519,
%T 2227,3263,4783,7009,10273,15055,22065,32337,47393,69457,101795,
%U 149187,218645,320439,469627,688271,1008711,1478337,2166609,3175319,4653657,6820265
%N a(0) = a(1) = a(2) = a(3) = 1; thereafter a(n) = a(n-2) + a(n-3) + a(n-4).
%C 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.
%H Michael De Vlieger, <a href="/A306276/b306276.txt">Table of n, a(n) for n = 0..6025</a>
%H Anthony Shannon, François Dubeau, Mine Uysal, and Engin Özkan, <a href="https://doi.org/10.7546/ijba.2022.26.4.000899">A Difference Equation Model of Infectious Disease</a>, Int. J. Bioautomation (2022) Vol. 26, No. 4, 339-352.
%F G.f.: (x^3 - x - 1)/(x^4 + x^3 + x^2 - 1).
%F a(n) = a(n-2) + a(n-3) + a(n-4) for n >= 4, a(n) = 1 for n < 4.
%F Lim_{n->infinity} a(n+1)/a(n) = A092526.
%t Nest[Append[#, Total@ #[[-4 ;; -2]] ] &, {1, 1, 1, 1}, 40] (* or *)
%t CoefficientList[Series[(x^3 - x - 1)/(x^4 + x^3 + x^2 - 1), {x, 0, 43}], x] (* _Michael De Vlieger_, Feb 09 2019 *)
%Y Cf. A000930, A092526.
%K nonn
%O 0,5
%A _Joseph Damico_, Feb 02 2019