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A005270 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.
(Formerly M1684)
3
1, 1, 1, 2, 6, 27, 177, 1680, 23009, 455368, 13067353, 546378617, 33472296082, 3021920660821, 404374532614122, 80646410554881100, 24095492607316134304, 10837141045948365696938, 7369252748590790186483284, 7606603491185739308318700818 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,4

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

REFERENCES

Fishburn, Peter C. and Roberts, Fred S.; Elementary sequences, sub-Fibonacci sequences. Discrete Appl. Math. 44 (1993), no. 1-3, 261-281.

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

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] <s[j] <= s[j-2]+s[j-1] for j>=3. - Emeric Deutsch, Feb 15 2007

EXAMPLE

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).

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

PROG

(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 */

CROSSREFS

Cf. A125051, A125052.

Cf. A005269.

Sequence in context: A070076 A227222 A130455 * A277611 A080839 A118085

Adjacent sequences:  A005267 A005268 A005269 * A005271 A005272 A005273

KEYWORD

nonn

AUTHOR

N. J. A. Sloane.

EXTENSIONS

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

Edited by Emeric Deutsch, Feb 15 2007

STATUS

approved

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Last modified October 20 15:43 EDT 2017. Contains 293625 sequences.