A005270 Number of sequences s of length n with s[1]=1, s[2]=1, s[j-1]=3.

1, 1, 1, 2, 6, 27, 177, 1680, 23009, 455368, 13067353, 546378617, 33472296082, 3021920660821, 404374532614122, 80646410554881100, 24095492607316134304, 10837141045948365696938, 7369252748590790186483284, 7606603491185739308318700818

Offset: 2

Author

N. J. A. Sloane

Keywords

• nonn

Comments

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

Examples

			G.f. = x^2 + x^3 + x^4 + 2*x^5 + 6*x^6 + 27*x^7 + 177*x^8 + 1680^x^9 + ...
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).
		

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Alois P. Heinz, Table of n, a(n) for n = 2..70
• Peter C. Fishburn and Fred S. Roberts, Elementary sequences, sub-Fibonacci sequences, Discrete Appl. Math. 44 (1993), no. 1-3, 261-281.

Crossrefs

Cf. A125051, A125052.

Cf. A005269.

Programs

Maple

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

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 */

Formula

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

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] =3. - Emeric Deutsch, Feb 15 2007

Extensions

a(12) from Paul D. Hanna, Nov 19 2006

Edited by Emeric Deutsch, Feb 15 2007