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A182699
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Number of emergent parts in all partitions of n.
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22
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0, 0, 0, 0, 1, 1, 4, 4, 10, 12, 22, 27, 47, 56, 89, 112, 164, 205, 294, 364, 505, 630, 845, 1052, 1393, 1719, 2235, 2762, 3533, 4343, 5506, 6730, 8443, 10296, 12786, 15531, 19161, 23161, 28374, 34201, 41621, 49975, 60513, 72385, 87200, 103999, 124670, 148209
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OFFSET
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0,7
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COMMENTS
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Here the "emergent parts" of the partitions of n are defined to be the parts (with multiplicity) of all the partitions that do not contain "1" as a part, removed by one copy of the smallest part of every partition. Note that these parts are located in the head of the outer shell of the partitions of n.
Also, here the "filler parts" of the partitions of n are defined to be the parts of the outer shell of the partitions of n that are not the emergent parts.
For n >= 4, length of row n of A183152. - Omar E. Pol, Aug 08 2011
Also total number of parts of the regions that do not contain 1 as a part in the outer shell of the partitions of n (Cf. A083751, A187219). - Omar E. Pol, Mar 04 2012
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LINKS
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Alois P. Heinz, Table of n, a(n) for n = 0..1000
O. E. Pol, Illustration: How to build the outer shell of the partitions (copy, paste and fill)
O. E. Pol, Illustration of the shell model of partitions (2D view)
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FORMULA
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a(n) = A138135(n) - A002865(n), n >= 1.
Contribution from Omar. E. Pol, Oct 21 2011 (Start):
a(n) = A006128(n) - A006128(n-1) - A000041(n), n >= 1.
a(n) = A138137(n) - A000041(n), n >= 1. (End)
a(n) = A076276(n) - A006128(n-1), n >= 1. - Omar E. Pol, Oct 30 2011
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EXAMPLE
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For n = 6 the partitions of 6 contain four "emergent" parts: (3), (4), (2), (2), so a(6) = 4. See below the location of the emergent parts.
6
(3) + 3
(4) + 2
(2) + (2) + 2
5 + 1
3 + 2 + 1
4 + 1 + 1
2 + 2 + 1 + 1
3 + 1 + 1 + 1
2 + 1 + 1 + 1 + 1
1 + 1 + 1 + 1 + 1 + 1
For a(10) = 22 see the link for the location of the 22 "emergent parts" (colored yellow and green) and the location of the 42 "filler parts" (colored blue) in the outer shell of the partitions of 10.
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MAPLE
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b:= proc(n, i) option remember; local t, h;
if n<0 then [0, 0, 0]
elif n=0 then [0, 1, 0]
elif i<2 then [0, 0, 0]
else t:= b(n, i-1); h:= b(n-i, i);
[t[1]+h[1]+h[2], t[2], t[3]+h[3]+h[1]]
fi
end:
a:= n-> b(n, n)[3]:
seq (a(n), n=0..50); # Alois P. Heinz, Oct 21 2011
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CROSSREFS
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Cf. A000041, A002865, A135010, A138121, A138135, A182700, A182709, A182740.
Sequence in context: A219939 A219471 A006477 * A058596 A180964 A209423
Adjacent sequences: A182696 A182697 A182698 * A182700 A182701 A182702
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KEYWORD
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nonn,easy
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AUTHOR
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Omar E. Pol, Nov 29 2010
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STATUS
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approved
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