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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.

%I A216053 #18 Feb 20 2020 20:41:10
%S A216053 2,3,5,8,13,20,31,46,68,98,140,196,273,374,509,685,916,1213,1598,2088,
%T A216053 2715,3507,4509,5764,7339,9297,11733,14743,18461,23026,28630,35472,
%U A216053 43821,53964,66274,81157,99134,120771,146786,177971,215309,259892,313066,376327
%N 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.
%H A216053 J. Stauduhar, <a href="/A216053/b216053.txt">Table of n, a(n) for n = 1..10000</a>
%F A216053 a(n) ~ exp(Pi*sqrt(2*n/3)) / (Pi*2^(3/2)*sqrt(n)). - _Vaclav Kotesovec_, May 24 2018
%F A216053 a(n) = A330661(2n,2) = A330661(2n+1,2). - _Alois P. Heinz_, Feb 20 2020
%e A216053 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.
%t A216053 RecurrenceTable[{a[n+1] == a[n] + PartitionsP[(n)], a[1] == 2}, a, {n, 1, 44}]
%Y A216053 A diagonal of A181187.
%Y A216053 Cf. A080577, A330661, A332706.
%K A216053 nonn
%O A216053 1,1
%A A216053 _J. Stauduhar_, Oct 12 2012