|
| |
|
|
A102848
|
|
Number of partitions of n into Fibonacci number of integer parts.
|
|
4
|
|
|
|
1, 1, 2, 3, 4, 6, 8, 10, 14, 18, 23, 29, 37, 47, 59, 74, 92, 114, 141, 173, 213, 261, 318, 387, 470, 569, 687, 827, 994, 1192, 1426, 1702, 2028, 2412, 2863, 3392, 4012, 4738, 5585, 6574, 7726, 9067, 10624, 12433, 14528, 16957, 19763, 23007, 26749, 31067, 36034
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,3
|
|
|
COMMENTS
|
A003107 & this sequence are different sequences. A003107 gives the number of partitions in which each part of n is a Fibonacci number, this sequence gives the number of partitions in which the number of parts is a Fibonacci number. Both sequences share the same values for the first 9 values. For example A003107(4) = 4 because of the following 4 partitions of 5: (3,1), (2,2), (2,1,1), (1,1,1,1) whereas a(4) is also 4 but because of different set of partitions: (4), (3,1), (2,2), (2,1,1).
|
|
|
LINKS
|
|
|
|
FORMULA
|
G.f.: 1 + Sum_{n>=2} x^Fibonacci(n)/Product_{i=1..Fibonacci(n)} (1-x^i). - Vladeta Jovovic, Mar 02 2005
|
|
|
EXAMPLE
|
a(5) = 6 since out of 7 possible partitions of 5 into integer parts, only 6 include a Fibonacci number of parts: (5), (4,1), (3,2), (3,1,1), (2,2,1), (1,1,1,1,1). The 7th integer partitions of 5 (2,1,1,1) is not counted since it includes 4 integer parts and 4 is not a Fibonacci number.
|
|
|
MAPLE
|
b:= proc(n, i, t) option remember; `if`(n=0 or i=1,
`if`((h-> issqr(h+4) or issqr(h-4))(5*(t+n)^2), 1, 0),
b(n, i-1, t) + b(n-i, min(i, n-i), t+1))
end:
a:= n-> b(n$2, 0):
|
|
|
MATHEMATICA
|
b[n_, i_, t_] := b[n, i, t] = If[n == 0 || i == 1, If[IntegerQ @ Sqrt[# + 4] || IntegerQ @ Sqrt[# - 4]&[5*(t + n)^2], 1, 0], b[n, i - 1, t] + b[n - i, Min[i, n - i], t + 1]];
a[n_] := b[n, n, 0];
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
easy,nonn
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
|
|
|
STATUS
|
approved
|
| |
|
|
