A194805 Number of parts that are visible in one of the three views of the section model of partitions version "tree" with n sections.

0, 1, 2, 4, 7, 11, 17, 25, 36, 51, 71, 97, 132, 177, 235, 310, 406, 527, 681, 874, 1116, 1418, 1793, 2256, 2829, 3532, 4393, 5445, 6727, 8282, 10168, 12445, 15190, 18491, 22452, 27192, 32859, 39613, 47651, 57199, 68522, 81920, 97756, 116434, 138435

Offset: 0

Author

Omar E. Pol, Jan 27 2012

Keywords

• nonn

Comments

The mentioned view of the section model looks like a tree (see example). Note that every column contains the same parts. For more information about the section model of partitions see A135010 and A194803.

Number of partitions of 2n-1 such that n-1 or n is a part, for n >=1. - Clark Kimberling, Mar 01 2014

Examples

			Illustration of one of the three views with seven sections:
.
.                   1
.                 2 1
.                   1 3
.                 2 1
.               4   1
.                   1 3
.                   1   5
.                 2 1
.               4   1
.             3     1
.           6       1
.                     3
.                       5
.                         4
.                           7
.
There are 25 parts that are visible, so a(7) = 25.
Using the formula we have a(7) = p(7) + p(7-1) - 1 = 15 + 11 - 1 = 25, where p(n) is the number of partitions of n.
		

Links

• Robert Price, Table of n, a(n) for n = 0..5000

Crossrefs

Cf. A000041, A000065, A084376, A135010, A138121, A141285, A194550, A194803, A194804.

Programs

Mathematica

Table[Count[IntegerPartitions[2 n - 1],  p_ /; Or[MemberQ[p, n - 1], MemberQ[p, n]]], {n, 50}]  (* Clark Kimberling, Mar 01 2014 *)
Table[PartitionsP[n] + PartitionsP[n-1] - 1, {n, 0, 44}] (* Robert Price, May 12 2020 *)

Formula

a(n) = A084376(n) - 1.

a(n) = A000041(n) + A000041(n-1) - 1, if n >= 1.

a(n) = A000041(n) + A000065(n-1), if n >= 1.