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A194805
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Number of parts that are visible in one of the three views of the section model of partitions version "tree" with n sections.
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10
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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
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OFFSET
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0,3
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COMMENTS
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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
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LINKS
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FORMULA
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EXAMPLE
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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
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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.
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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