A286041 Number of partitions of n with parts differing by at least two and with an even number of parts.

1, 0, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 8, 8, 10, 11, 14, 15, 19, 21, 26, 29, 35, 39, 47, 52, 61, 68, 79, 87, 100, 110, 126, 138, 156, 171, 193, 211, 237, 259, 290, 317, 354, 387, 432, 472, 525, 575, 639, 699, 776, 849, 941, 1030, 1139, 1246, 1377, 1505, 1659, 1813, 1996, 2178, 2394, 2610, 2863, 3119, 3415, 3716, 4064, 4416, 4820, 5234, 5705, 6187, 6735, 7297, 7932

Offset

0,7

Links

Alois P. Heinz, Table of n, a(n) for n = 0..5000

Formula

a(n) = A003114(n) - A286316(n).

a(n) ~ phi^(1/2) * exp(2*Pi*sqrt(n/15)) / (4 * 3^(1/4) * 5^(1/2) * n^(3/4)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Mar 05 2020

Mathematica

Table[Length@ Select[IntegerPartitions@n, Min[-Differences@#] > 1  && EvenQ@Length@# &], {n, 80}]

Crossrefs

Cf. A003114, A286316, A039924.

Sequence in context: A026828 A025153 A026803 * A027192 A194255 A194245

Adjacent sequences: A286038 A286039 A286040 * A286042 A286043 A286044

Keyword

nonn

Author

George Beck, May 06 2017

Extensions

a(0)=1 prepended by Alois P. Heinz, May 07 2017

Status

approved