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A207034
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Sum of all parts minus the number of parts in the n-th zone of the shell model of partitions.
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6
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0, 1, 2, 2, 3, 3, 4, 3, 4, 4, 5, 4, 5, 5, 6, 4, 5, 5, 6, 6, 6, 7, 5, 6, 6, 7, 6, 7, 7, 8, 5, 6, 6, 7, 7, 7, 8, 7, 8, 8, 8, 9, 6, 7, 7, 8, 7, 8, 8, 9, 8, 8, 9, 9, 9, 10, 6, 7, 7, 8, 8, 8, 9, 8, 9, 9, 9, 10, 8, 9, 9, 10, 9, 10, 10, 10, 11, 7, 8, 8, 9, 8, 9
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OFFSET
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1,3
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COMMENTS
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a(n) is also the column number in which is located the part of size 1 in the n-th zone of the tail of the outer shell of the partitions of k, minus the column number in which is located the part of size 1 in the first zone of the same tail, when k -> oo.
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LINKS
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Alois P. Heinz, Table of n, a(n) for n = 1..10143
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FORMULA
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a(n) = t(n) - A194548(n), if n >= 2, where t(n) is the n-th element of the following sequence: triangle read by rows in which row n lists n repeated k times, where k = A187219(n). - Omar E. Pol, Apr 12 2012
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EXAMPLE
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Illustration of initial terms, n = 1..15. Consider the last 15 zones of the tail of the outer shell of the partitions of any integer >= 8. The tail contains at least A000041(8-1) = 15 parts of size 1. a(n) is also the number of dots in zone n.
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n Tail a(n)
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15 1 . . . . . . 6
14 1 . . . . . 5
13 1 . . . . . 5
12 1 . . . . 4
11 1 . . . . . 5
10 1 . . . . 4
9 1 . . . . 4
8 1 . . . 3
7 1 . . . . 4
6 1 . . . 3
5 1 . . . 3
4 1 . . 2
3 1 . . 2
2 1 . 1
1 1 0
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Written as a triangle:
0;
1;
2;
2,3;
3,4;
3,4,4,5;
4,5,5,6;
4,5,5,6,6,6,7;
5,6,6,7,6,7,7,8;
5,6,6,7,7,7,8,7,8,8,8,9;
6,7,7,8,7,8,8,9,8,8,9,9,9,10;
6,7,7,8,8,8,9,8,9,9,9,10,8,9,9,10,9,10,10,10,11;
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CROSSREFS
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Row r has length A187219(r). Partial sums give A207038. Row sums give A207035. Right border gives A001477. Where records occur give A000041 without repetitions.
Cf. A002865, A135010, A138121, A182703, A194548, A196087, A207031, A207032, A207035.
Sequence in context: A117498 A064097 A014701 * A218829 A056239 A161511
Adjacent sequences: A207031 A207032 A207033 * A207035 A207036 A207037
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KEYWORD
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nonn,tabf
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AUTHOR
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Omar E. Pol, Feb 20 2012
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STATUS
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approved
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