A216053 a(n) is the position of the last two-tuple within the reverse lexicographic set of partitions of 2n and 2n+1, with a(1)-a(n) representing the positions of every 2-tuple partition of 2n and 2n+1.

2, 3, 5, 8, 13, 20, 31, 46, 68, 98, 140, 196, 273, 374, 509, 685, 916, 1213, 1598, 2088, 2715, 3507, 4509, 5764, 7339, 9297, 11733, 14743, 18461, 23026, 28630, 35472, 43821, 53964, 66274, 81157, 99134, 120771, 146786, 177971, 215309, 259892, 313066, 376327

Offset: 1

Author

J. Stauduhar, Oct 12 2012

Keywords

• nonn

Examples

			With n = 3, 2n = 6.  The partitions of 6 are {{6}, {5,1}, {4,2}, {4,1,1}, {3,3}, {3,2,1}, {3,1,1,1}, {2,2,2}, {2,2,1,1}, {2,1,1,1,1}, {1,1,1,1,1,1}}.  The last 2-tuple is located at position 5. The positions of all 2-tuples are 2, 3, and 5.
		

Links

• J. Stauduhar, Table of n, a(n) for n = 1..10000

Crossrefs

A diagonal of A181187.

Cf. A080577, A330661, A332706.

Programs

Mathematica

RecurrenceTable[{a[n+1] == a[n] + PartitionsP[(n)], a[1] == 2}, a, {n, 1, 44}]

Formula

a(n) ~ exp(Pi*sqrt(2*n/3)) / (Pi*2^(3/2)*sqrt(n)). - Vaclav Kotesovec, May 24 2018

a(n) = A330661(2n,2) = A330661(2n+1,2). - Alois P. Heinz, Feb 20 2020