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A194805 Number of parts that are visible in one of the three views of the section model of partitions version "tree" with n sections. 10
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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

LINKS

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

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.

EXAMPLE

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.

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 *)

CROSSREFS

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

Sequence in context: A096914 A004250 A289060 * A084842 A289177 A249039

Adjacent sequences:  A194802 A194803 A194804 * A194806 A194807 A194808

KEYWORD

nonn

AUTHOR

Omar E. Pol, Jan 27 2012

STATUS

approved

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Last modified July 4 18:38 EDT 2020. Contains 335448 sequences. (Running on oeis4.)