%I #6 Mar 03 2020 09:58:29
%S 1,3,9,36,228,2196,33901,862503,36346723,2564238411,304902857694,
%T 61384367733677,21020435566780278,12292402317454051941,
%U 12319906894146608845054,21234027294331775378957366
%N Number of sequences s of length n, with s[1]=1, s[2]=1, s[3]=1, s[k-1] <=s[k] <= s[k-1]+s[k-2]+s[k-3] (s is called a sub-tribonacci sequence of length n).
%H Peter C. Fishburn and Fred S. Roberts, <a href="https://doi.org/10.1016/0166-218X(93)90236-H">Elementary sequences, sub-Fibonacci sequences</a>, Discrete Appl. Math. 44 (1993), no. 1-3, 261-281.
%F See the Maple program; f[k](x,y,z) is the number of sequences s[1], s[2], ..., s[k+3] such that s[1]=x, s[2]=y, s[3]=z, s[j-1] <=s[j] <= s[j-3]+s[j-2]+s[j-1].
%e a(5)=9 because we have (1,1,1,1,1), (1,1,1,1,2), (1,1,1,1,3), (1,1,1,2,2), (1,1,1,2,3), (1,1,1,2,4), (1,1,1,3,3), (1,1,1,3,4), (1,1,1,3,5).
%p f[0]:=1:for k from 0 to 20 do f[k+1]:=factor(sum(subs({x=y, y=z, z=u}, f[k]), u=z..x+y+z)) od: seq(subs({x=1, y=1,z=1}, f[k]), k=0..20);
%Y Cf. A005269.
%K nonn
%O 3,2
%A _Emeric Deutsch_, Feb 14 2007