|
|
A005269
|
|
a(n) = number of length-n sequences s with s[1]=1, s[2]=1, s[k-1] <=s[k] <= s[k-2]+s[k-1] (s is called a sub-Fibonacci sequence of length n).
(Formerly M1234)
|
|
9
|
|
|
1, 2, 4, 10, 31, 127, 711, 5621, 64049, 1067599, 26287664, 963023487, 52766766100, 4342736509018, 538755914902622, 101067429677072459, 28751803102222498512, 12436935036300286507123, 8200693250120852291693833, 8262592110164298068793701546
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
2,2
|
|
REFERENCES
|
Fishburn, Peter C.; Roberts, Fred S., Uniqueness in finite measurement. Applications of combinatorics and graph theory to the biological and social sciences, 103--137, IMA Vol. Math. Appl., 17, Springer, New York, 1989. MR1009374 (90e:92099)
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
|
|
LINKS
|
Peter C. Fishburn and Fred S. Roberts, Uniqueness in finite measurement, in Applications of combinatorics and graph theory to the biological and social sciences, 103--137, IMA Vol. Math. Appl., 17, Springer, New York, 1989. MR1009374 (90e:92099). [Annotated scan of five pages only]
|
|
FORMULA
|
See the Maple program; f[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]. - Emeric Deutsch and Don Reble, Feb 07 2005
|
|
EXAMPLE
|
G.f. = x^2 + 2*x^3 + 4*x^4 + 10*x^5 + 31*x^6 + 127*x^7 + 711*x^8 + 5621*x^9 + ...
a(4)=4 because we have (1,1,1,1), (1,1,1,2), (1,1,2,2), (1,1,2,3).
|
|
MAPLE
|
f[0]:=1:for k from 0 to 19 do f[k+1]:=expand(sum(subs({x=y, y=z}, f[k]), z=y..x+y)) od: seq(subs({x=1, y=1}, f[k]), k=0..19);
|
|
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
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|