A121659 Number of partitions of n into parts with at most one part not greater than 2.

1, 1, 1, 2, 3, 4, 5, 7, 9, 12, 15, 20, 25, 32, 40, 51, 63, 79, 97, 121, 148, 182, 221, 271, 328, 398, 479, 579, 694, 834, 995, 1190, 1415, 1684, 1995, 2366, 2793, 3298, 3881, 4569, 5360, 6288, 7355, 8603, 10037, 11705, 13619, 15842, 18388, 21333, 24703

Offset: 1

Author

Reinhard Zumkeller, Aug 14 2006

Keywords

• nonn

Comments

The number of partitions of n that have the first part at least thrice larger than the second part. Example: a(8) = #{8, 7+1, 6+2, 6+1+1, 5+1+1+1, 4+1+1+1+1, 3+1+1+1+1+1}=7. [Mircea Merca, Jul 24 2011]

Examples

			a(8) = #{8,7+1,6+2,5+3,4+4,4+3+1,3+3+2} = 7;
a(9) = #{9,8+1,7+2,6+3,5+4,5+3+1,4+4+1,4+3+2,3+3+3} = 9.
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
• Mircea Merca, Fast algorithm for generating ascending compositions, arXiv:1903.10797 [math.CO], 2019.

Crossrefs

Cf. A027336, A008483, A121081, A000041.

Formula

a(n) = A121081(n) - A008483(n-3) for n>2.

a(n) = p(n) - p(n-2) - p(n-3) + p(n-5) where p(n) = A000041(n). See Merca p. 6. - Michel Marcus, Mar 27 2019

a(n) ~ exp(Pi*sqrt(2*n/3)) * Pi^2 / (4*sqrt(3)*n^2) * (1 - (18/Pi + 61*Pi/24)/sqrt(6*n)). - Vaclav Kotesovec, Jan 15 2022