A000094 -id:A000094 - cp's OEIS frontend

A115982 Number of planar partitions that are not corners.

0, 0, 0, 1, 3, 10, 23, 54, 112, 228, 437, 826, 1499, 2685, 4688, 8079, 13668, 22875, 37738, 61676, 99672, 159742, 253681, 399962, 625741, 972756, 1502302, 2306988, 3522492, 5351239, 8088469, 12170163, 18229411, 27192571

Offset: 1

Alford Arnold, Feb 17 2006

a(n) can also be approximated by considering A000094 since A000094(n) = A000041(n) - n = 0 0 0 1 2 5 8 14 21 32 ... with partial sums 0 0 0 1 3 8 16 30 51 83 ... which counts many of the initial cases. The remaining cases form 0 0 0 0 0 2 7 24 ... counting for n=6, 22/11 and 21/21.

			The planar partitions begin 1 3 6 13 24 48 ... A000219 with corners 1 3 6 12 21 38 ... A006330; therefore the present sequence begins 0 0 0 1 3 10 ...

Cf. A000219, A006330, A000041, A000094.

a(n) = A000219(n) - A006330(n)

Edited with additional terms by Franklin T. Adams-Watters, Mar 10 2006

A182377 Total sum of positive ranks of all regions in the last shell of n.

Original entry on oeis.org

0, 0, 0, 1, 2, 5, 8, 14, 21, 32, 45, 67, 91

Offset: 1

Omar E. Pol, Apr 29 2012

The rank of a region of n is the largest part minus the number of parts. For the definition of "region of n" see A206437. For the definition of "last shell of n" see A135010.

a(n) is also the sum of positive integers in row n of triangle A194447. First differs from A000094 at a(12).

			For n = 7 the last shell of 7 contains four regions: [3],[5,2],[4],[7,3,2,2,1,1,1,1,1,1,1,1,1,1,1] so we have:
----------------------------------------------------------
.        Largest    Number
Region     part    of parts    Rank
----------------------------------------------------------
.  1        3         1          2
.  2        5         2          3
.  3        4         1          3
.  4        7        15         -8
.
The sum of positive ranks is a(7) = 2 + 3 + 3 = 8.

Omar E. Pol, Illustration of the seven regions of 5

Cf. A000094, A135010, A138121, A182699, A182709, A183152, A186114, A187219, A194436-A194439, A194447-A194448, A196025, A206437, A209616.

A194451 Partition numbers of positive integers and positive integers interleaved.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 5, 4, 7, 5, 11, 6, 15, 7, 22, 8, 30, 9, 42, 10, 56, 11, 77, 12, 101, 13, 135, 14, 176, 15, 231, 16, 297, 17, 385, 18, 490, 19, 627, 20, 792, 21, 1002, 22, 1255, 23, 1575, 24, 1958, 25, 2436, 26, 3010, 27, 3718, 28, 4565, 29, 5604, 30

Offset: 1

Omar E. Pol, Nov 01 2011

nonn

First differences of A194450. The length of the edges in a rectangular spiral whose vertices are the numbers A194450. The spiral contains exactly between its edges the successive last sections of the partitions of the natural numbers. For more information see A135010 and A138121.

Cf. A000041, A194450.

a(2n-1) = p(n), a(2n) = n.
a(2n-1) = A000041(n), a(2n) = A000027(n).
a(2n-1) + a(2n) = A133041(n).
a(2n-1) - a(2n) = A000094(n+1).
a(2n-1) * a(2n) = A066186(n).

A364451 a(n) is the number of trees of diameter 4 with n vertices that are N-games in peg duotaire.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 5, 7, 10, 13, 18, 22, 29, 34, 42, 49, 60, 69, 86, 100, 121, 139, 164, 187, 219, 252, 296, 343, 400, 458, 532, 605, 696, 794, 917, 1050, 1214, 1389, 1599, 1823, 2087, 2371, 2710, 3080, 3521

Offset: 1

Michael Carrion, Nathan Hurtig, Maggie X. Lai, Sarah Lohrey, Brittany Ohlinger, Jul 25 2023

nonn

Peg duotaire is an impartial normal-play two-player game played on a simple graph, in which each vertex starts with a peg in it. If all vertices have a peg (i.e. the first turn), a move consists of removing some peg from a vertex.
If some vertex does not have a peg, then a move hops one peg over another, landing in an adjacent hole and removing the jumped peg. Formally, it is three vertices x, y, z where x, y are adjacent and y, z are adjacent, and x, y have pegs and z does not. After the move, x, y do not have pegs and z does.
Note than this sequence is always less than or equal to the number of trees of diameter 4 with n vertices, see A000094.

			There is only one tree of diameter 4 with 5 vertices. It is an N-game, as evidenced by the below winning strategy for the first player. We use 1 to represent a vertex with a peg and 0 otherwise.
   1-1-1-1-1
       |
   1-0-1-1-1
       |     (move is forced)
   1-1-0-0-1
       |
   0-0-1-0-1 (no moves remain)

E. R. Berlekamp, J. H. Conway, and R. K. Guy, Winning Ways for Your Mathematical Plays. Vol. 1, CRC Press, 2001.

R. A. Beeler and A. D. Gray, An Introduction to Peg Duotaire on Graphs, Journal of Combinatorial Mathematics and Combinatorial Computing, Vol. 104, 2018, pp. 171-186.
Nathan Hurtig, SageMath program to compute values

Cf. A000094.

a(n) <= A000094(n).

cp's OEIS Frontend

A115982 Number of planar partitions that are not corners.

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A182377 Total sum of positive ranks of all regions in the last shell of n.

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A194451 Partition numbers of positive integers and positive integers interleaved.

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A364451 a(n) is the number of trees of diameter 4 with n vertices that are N-games in peg duotaire.

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