A006726 -id:A006726 - cp's OEIS frontend

A008855 Triangle T(n,k), n>=1, read by rows, where T(n,k) is the number of lattice polygons with area n and perimeter 2*k.

1, 2, 6, 1, 18, 8, 55, 2, 40, 174, 22, 168, 566, 6, 134, 676, 1868, 1, 72, 656, 2672, 6237, 30, 482, 2992, 10376, 21050, 8, 310, 2592, 13160, 39824, 71666, 2, 151, 2086, 12862, 56162, 151878, 245696, 68, 1392, 11717, 61032, 234520, 576656, 847317, 22, 864, 9332, 60864, 279492, 965136, 2181496, 2937116

Offset: 1

N. J. A. Sloane

Row lengths are 1, 1, 1, 2, 2, 3, 3, 4, 5, 5, ...

			T(1,2) = 1;
   +--+
   |  |
   +--+
T(2,3) = 2;
   +--+
   |  |
   +--+   +--+--+
   |  |   |  |  |
   +--+   +--+--+
T(3,4) = 6;
   +--+
   |  |
   +--+                +--+
   |  |                |  |
   +--+   +--+--+--+   +--+--+
   |  |   |  |  |  |   |  |  |
   +--+   +--+--+--+   +--+--+
      +--+   +--+--+   +--+--+
      |  |   |  |  |   |  |  |
   +--+--+   +--+--+   +--+--+
   |  |  |   |  |         |  |
   +--+--+   +--+         +--+
============================================
n\k | 2  3  4   5   6    7    8     9    10
----|---------------------------------------
  1 | 1;
  2 |    2;
  3 |       6;
  4 |       1, 18;
  5 |           8, 55;
  6 |           2, 40, 174;
  7 |              22, 168, 566;
  8 |               6, 134, 676, 1868;
  9 |               1,  72, 656, 2672, 6237;

Andrey Zabolotskiy, Table of n, a(n) for n = 1..120 (rows 1..20, from Enting & Guttmann)
I. G. Enting and A. J. Guttmann, On the area of square lattice polygons, J. Statist. Phys., 58 (1990), 475-484.

Cf. A006725, A006726, A259857.

More terms from Sean A. Irvine, Apr 12 2018

Examples added and name edited by Seiichi Manyama, Apr 04 2020

A259857 Triangle T(n,k), n>=1, 2<=k<=n+1, read by rows, where T(n,k) is the number of self-avoiding square-lattice polygons by area n and perimeter 2*k.

Original entry on oeis.org

1, 0, 2, 0, 0, 6, 0, 0, 1, 18, 0, 0, 0, 8, 55, 0, 0, 0, 2, 40, 174, 0, 0, 0, 0, 22, 168, 566, 0, 0, 0, 0, 6, 134, 676, 1868, 0, 0, 0, 0, 1, 72, 656, 2672, 6237, 0, 0, 0, 0, 0, 30, 482, 2992, 10376, 21050, 0, 0, 0, 0, 0, 8, 310, 2592, 13160, 39824, 71666, 0, 0, 0, 0, 0, 2, 151, 2086, 12862, 56162, 151878, 245696

Offset: 1

N. J. A. Sloane, Jul 07 2015

			Triangle begins:
==========================================================
n\k  | 2 3 4  5  6   7   8    9    10    11     12     13
-----|----------------------------------------------------
   1 | 1,
   2 | 0,2,
   3 | 0,0,6,
   4 | 0,0,1,18
   5 | 0,0,0, 8,55,
   6 | 0,0,0, 2,40,174,
   7 | 0,0,0, 0,22,168,566,
   8 | 0,0,0, 0, 6,134,676,1868,
   9 | 0,0,0, 0, 1, 72,656,2672, 6237,
  10 | 0,0,0, 0, 0, 30,482,2992,10376,21050,
  11 | 0,0,0, 0, 0,  8,310,2592,13160,39824, 71666,
  12 | 0,0,0, 0, 0,  2,151,2086,12862,56162,151878,245696,

I. G. Enting and A. J. Guttmann, On the area of square lattice polygons, J. Statist. Phys., 58 (1990), 475-484. See Table 1.

A006725 and A006726 are diagonals.
Row sums give A006724.
Cf. A008855 (with 0 omitted).

a(7)-a(10) inserted by Seiichi Manyama, Apr 04 2020

cp's OEIS Frontend

A008855 Triangle T(n,k), n>=1, read by rows, where T(n,k) is the number of lattice polygons with area n and perimeter 2*k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions