cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A351994 Number of spanning trees in a hexagon of size n in the triangular grid.

Original entry on oeis.org

1, 320, 2300606464, 289899537900576358400, 614482906548854364363387716704247808, 21564742087547836976004856537464240189331001616154755072, 12433415382338420812828401445037903120443542018197863908895102595928462876835840
Offset: 0

Views

Author

Peter Kagey, Feb 28 2022

Keywords

Comments

The hexagon of size n in the triangular grid has A003215(n) vertices.

Links

Crossrefs

Cf. A007341 (square in square grid), A116469 (rectangle in square grid), A174579 (triangle in triangular grid), A351888 (triangle in hexagonal grid), A352022 (hexagon in hexagonal grid).

A352022 Number of spanning trees in a hexagon of size n in the hexagonal grid.

Original entry on oeis.org

1, 6, 176400, 95437674624600, 878617506040998925900403712, 134527385723138237635420920683683500322908000, 339161155484890894029987276076070590877762998258747782208794132480, 14004953513181662639884345044013838519837158205213642081126147144590500534440163767670000000
Offset: 0

Views

Author

Peter Kagey, Feb 28 2022

Keywords

Comments

The hexagon of size n in the hexagonal grid has A033581(n) = 6*n^2 vertices.

Links

Crossrefs

Cf. A007341 (square in square grid), A116469 (rectangle in square grid), A174579 (triangle in triangular grid), A351888 (triangle in hexagonal grid), A351994 (hexagon in triangular grid).

A360062 Triangle read by rows: T(m,n) is the number of spanning trees in the graph whose nodes are the integer lattice points (x,y) with 0 <= x < m and 0 <= y < n, and with an edge between two nodes if there is no other integer lattice point on the line segment between them; 1 <= n <= m.

Original entry on oeis.org

1, 1, 16, 1, 576, 496125, 1, 41616, 1830420480, 375297659043840, 1, 5085025, 10361547386325, 166557643451782840320, 5885897714143664700439342125, 1, 945193536, 144188666818560000, 258848560805325726352932864, 1192037309255692352595217996892160000, 36939045170346949681155330481716034613142893328
Offset: 1

Views

Author

Pontus von Brömssen, Jan 24 2023

Keywords

Examples

			Triangle begins:
  m\n| 1     2          3               4
  ---+-----------------------------------
  1  | 1
  2  | 1    16
  3  | 1   576     496125
  4  | 1 41616 1830420480 375297659043840

Links

Crossrefs

Cf. A116469, A247943, A360063.

A338832 Number of spanning trees in the k_1 X ... X k_j grid graph, where (k_1 - 1, ..., k_j - 1) is the partition with Heinz number n.

Original entry on oeis.org

1, 1, 1, 4, 1, 15, 1, 384, 192, 56, 1, 31500, 1, 209, 2415, 42467328, 1, 49766400, 1, 2558976, 30305, 780, 1, 3500658000000, 100352, 2911, 8193540096000, 207746836, 1, 76752081000, 1, 20776019874734407680, 380160, 10864, 4140081, 242716067758080000000, 1
Offset: 1

Views

Author

Pontus von Brömssen, Nov 11 2020

Keywords

Comments

a(n) > 1 precisely when n is composite.

Examples

			The partition (2, 2, 1) has Heinz number 18 and the 3 X 3 X 2 grid graph has a(18) = 49766400 spanning trees.

Links

Crossrefs

2 X n grid: A001353(n) = a(2*prime(n-1))
3 X n grid: A006238(n) = a(3*prime(n-1))
4 X n grid: A003696(n) = a(5*prime(n-1))
5 X n grid: A003779(n) = a(7*prime(n-1))
6 X n grid: A139400(n) = a(11*prime(n-1))
7 X n grid: A334002(n) = a(13*prime(n-1))
8 X n grid: A334003(n) = a(17*prime(n-1))
9 X n grid: A334004(n) = a(19*prime(n-1))
10 X n grid: A334005(n) = a(23*prime(n-1))
n X n grid: A007341(n) = a(prime(n-1)^2)
m X n grid: A116469(m,n) = a(prime(m-1)*prime(n-1))
2 X 2 X n grid: A003753(n) = a(4*prime(n-1))
2 X n X n grid: A067518(n) = a(2*prime(n-1)^2)
n X n X n grid: A071763(n) = a(prime(n-1)^3)
2 X ... X 2 grid: A006237(n) = a(2^n)

Formula

a(n) = Product_{n_1=0..k_1-1, ..., n_j=0..k_j-1; not all n_i=0} Sum_{i=1..j} (2*(1 - cos(n_i*Pi/k_i))) / Product_{i=1..j} k_i, where (k_1 - 1, ..., k_j - 1) is the partition with Heinz number n.

A360922 Array read by antidiagonals: T(m,n) is the number of acyclic orientations in the grid graph P_m X P_n.

Original entry on oeis.org

1, 2, 2, 4, 14, 4, 8, 98, 98, 8, 16, 686, 2398, 686, 16, 32, 4802, 58670, 58670, 4802, 32, 64, 33614, 1435414, 5015972, 1435414, 33614, 64, 128, 235298, 35118638, 428816558, 428816558, 35118638, 235298, 128, 256, 1647086, 859207558, 36659327366, 128091434266, 36659327366, 859207558, 1647086, 256
Offset: 1

Views

Author

Andrew Howroyd, Mar 07 2023

Keywords

Examples

			Array begins:
=====================================================
m\n|  1     2        3           4              5 ...
---+-------------------------------------------------
1  |  1     2        4           8             16 ...
2  |  2    14       98         686           4802 ...
3  |  4    98     2398       58670        1435414 ...
4  |  8   686    58670     5015972      428816558 ...
5  | 16  4802  1435414   428816558   128091434266 ...
6  | 32 33614 35118638 36659327366 38261306901842 ...
  ...

Links

Crossrefs

Main diagonal is A080690.
Rows 1..2 are A000079(n-1), A109808.
Cf. A116469 (spanning trees), A178435, A207868 (unlabeled colorings).

Formula

T(m,n) = T(n,m).
Previous Showing 21-25 of 25 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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