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%I M1684 #23 Mar 03 2020 09:59:42
%S 1,1,1,2,6,27,177,1680,23009,455368,13067353,546378617,33472296082,
%T 3021920660821,404374532614122,80646410554881100,24095492607316134304,
%U 10837141045948365696938,7369252748590790186483284,7606603491185739308318700818
%N Number of sequences s of length n with s[1]=1, s[2]=1, s[j-1]<s[j]<=s[j-2]+s[j-1] for j>=3.
%C The sequences of length n that are counted here are sub-Fibonacci sequences (A005269) with the property that its members, except for the initial two terms, strictly increase. - _Emeric Deutsch_, Feb 15 2007
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Alois P. Heinz, <a href="/A005270/b005270.txt">Table of n, a(n) for n = 2..70</a>
%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 a(n) equals the number of nodes in generation n-2 of the sub-Fibonacci tree (A125051) for n>=2. - _Paul D. Hanna_, Nov 19 2006
%F See the Maple program; g[k](x, y) is the number of sequences s[1], s[2], ..., s[k+2] such that s[1]=x, s[2]=y, s[j-1] <s[j] <= s[j-2]+s[j-1] for j>=3. - _Emeric Deutsch_, Feb 15 2007
%e G.f. = x^2 + x^3 + x^4 + 2*x^5 + 6*x^6 + 27*x^7 + 177*x^8 + 1680^x^9 + ...
%e a(2)=6 because we have (1,1,2,3,4,5), (1,1,2,3,4,6), (1,1,2,3,4,7), (1,1,2,3,5,6), (1,1,2,3,5,7) and (1,1,2,3,5,8).
%p g[0]:=1:for k from 0 to 20 do g[k+1]:=expand(sum(subs({x=y, y=z}, g[k]), z=y+1..x+y)) od:seq(subs({x=1, y=1}, g[k]), k=0..20); # _Emeric Deutsch_, Feb 15 2007
%o (PARI) {a(n) = if(n<2, return(0)); my(c, e); forvec(s=vector(n, i, [1, fibonacci(i)]), e=0; for(k=3, n, if( s[k-1]>=s[k] || s[k]>s[k-2]+s[k-1], e=1; break)); if(e, next); c++, 1); c}; /* _Michael Somos_, Dec 02 2016 */
%Y Cf. A125051, A125052.
%Y Cf. A005269.
%K nonn
%O 2,4
%A _N. J. A. Sloane_.
%E a(12) from _Paul D. Hanna_, Nov 19 2006
%E Edited by _Emeric Deutsch_, Feb 15 2007
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