OFFSET
1,2
COMMENTS
The Fibonacci sequence starts with two 1's, that are to be considered distinct (see illustration).
LINKS
Paul Tek, Table of n, a(n) for n = 1..10000
Paul Tek, illustration of the first terms
EXAMPLE
The first (positive) Fibonacci numbers are: 1, 1, 2, 3, 5, 8...
The number 5 can be partitioned into:
- 1 consecutive initial Fibonacci number in 1 way: 5*1,
- 2 consecutive initial Fibonacci numbers in 4 ways: 4*1 + 1*1, 3*1 + 2*1, 2*1 + 3*1, 1*1 + 4*1,
- 3 consecutive initial Fibonacci numbers in 2 way: 2*1 + 1*1 + 1*2, 1*1 + 2*1 + 1*2.
Hence, a(5) = 1+4+2 = 7.
MAPLE
with(combinat):
b:= proc(n, i) option remember; local f; f:= fibonacci(i);
`if`(n=0 or i=1, 1, `if`(i<1 , 0, `if`(i=2, n+1,
b(n, i-1)+`if`(f>n, 0, b(n-f, i)))))
end:
a:= proc(n) local i, m, s; m, s:= n, 0;
for i do m:= m-fibonacci(i);
if m<0 then break fi;
s:= s+ b(m, i)
od; s
end:
seq(a(n), n=1..100); # Alois P. Heinz, Apr 29 2013
MATHEMATICA
b[n_, i_] := b[n, i] = Module[{f = Fibonacci[i]}, If[n==0 || i==1, 1, If[i < 1, 0, If[i==2, n + 1, b[n, i - 1] + If[f>n, 0, b[n - f, i]]]]]];
a[n_] := Module[{i, m = n, s = 0}, 0; For[i = 1, True, i++, m = m - Fibonacci[i]; If[m<0, Break[]]; s = s + b[m, i]]; s];
Array[a, 100] (* Jean-François Alcover, Nov 20 2020, after Alois P. Heinz *)
PROG
(PARI)
a(n) = {
my(s=0, P=1, k=1, x='x);
while(P!=0,
s=s+polcoeff(P, n);
P=(P*sum(z=1, n/fibonacci(k), x^(fibonacci(k)*z)))+O(x^(n+1));
k=k+1
);
return(s); } /* Paul Tek, Apr 28 2013 */
CROSSREFS
KEYWORD
nonn
AUTHOR
Frank Schwellinger (nummer_eins(AT)web.de), Feb 05 2010
EXTENSIONS
Corrected and extended by Paul Tek, Apr 28 2013
STATUS
approved
