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A172491 Number of partitions of n into consecutive initial Fibonacci numbers. 1
1, 2, 3, 5, 7, 10, 14, 19, 25, 33, 42, 54, 68, 85, 105, 129, 157, 190, 228, 273, 324, 384, 452, 530, 619, 720, 834, 964, 1109, 1273, 1456, 1661, 1890, 2145, 2428, 2743, 3091, 3477, 3902, 4371, 4887, 5454, 6076, 6758, 7503, 8319, 9208, 10178, 11234, 12382 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
The Fibonacci sequence starts with two 1's, that are to be considered distinct (see illustration).
LINKS
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
Cf. A000045 (Fibonacci numbers).
Cf. A000009.
Sequence in context: A304712 A175842 A008581 * A036469 A238658 A116480
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

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Last modified May 24 23:50 EDT 2024. Contains 372782 sequences. (Running on oeis4.)