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.

Showing 1-5 of 5 results.

A378932 Array read by antidiagonals: T(m,n) is the number of minimal edge cuts in the grid graph P_m X P_n.

Original entry on oeis.org

0, 1, 1, 2, 6, 2, 3, 15, 15, 3, 4, 28, 53, 28, 4, 5, 45, 146, 146, 45, 5, 6, 66, 356, 627, 356, 66, 6, 7, 91, 809, 2471, 2471, 809, 91, 7, 8, 120, 1759, 9292, 16213, 9292, 1759, 120, 8, 9, 153, 3716, 33878, 103196, 103196, 33878, 3716, 153, 9, 10, 190, 7702, 120771, 642364, 1123743, 642364, 120771, 7702, 190, 10
Offset: 1

Views

Author

Andrew Howroyd, Dec 11 2024

Keywords

Comments

T(m,n) is the number of partitionings of an m X n checkerboard into two edgewise-connected sets.

Examples

			Table starts:
===================================================
m\n | 1  2    3     4      5        6         7 ...
----+----------------------------------------------
  1 | 0  1    2     3      4        5         6 ...
  2 | 1  6   15    28     45       66        91 ...
  3 | 2 15   53   146    356      809      1759 ...
  4 | 3 28  146   627   2471     9292     33878 ...
  5 | 4 45  356  2471  16213   103196    642364 ...
  6 | 5 66  809  9292 103196  1123743  12028981 ...
  7 | 6 91 1759 33878 642364 12028981 221984391 ...
  ...

Links

Crossrefs

Main diagonal is A068416.
Rows 1..4 are A001477(n-1), A000384, A378933, A378934.
Rows 3..8 multiplied by 2 are A166761, A166766, A166769, A166771, A166773, A166774.
Cf. A287151, A359990.

Formula

T(m,n) = T(n,m).

A359993 Array read by antidiagonals: T(m,n) is the number of connected spanning subgraphs in the grid graph P_m X P_n.

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 23, 23, 1, 1, 105, 431, 105, 1, 1, 479, 7857, 7857, 479, 1, 1, 2185, 142625, 555195, 142625, 2185, 1, 1, 9967, 2587279, 38757695, 38757695, 2587279, 9967, 1, 1, 45465, 46929343, 2698167665, 10286937043, 2698167665, 46929343, 45465, 1
Offset: 1

Views

Author

Andrew Howroyd, Jan 28 2023

Keywords

Comments

Also T(m,n) except when m = n = 0 is the number of connected edge covers in the m X n grid graph.

Examples

			Table starts:
=================================================================
m\n| 1    2       3          4             5                6
---+-------------------------------------------------------------
1  | 1    1       1          1             1                1 ...
2  | 1    5      23        105           479             2185 ...
3  | 1   23     431       7857        142625          2587279 ...
4  | 1  105    7857     555195      38757695       2698167665 ...
5  | 1  479  142625   38757695   10286937043    2711895924889 ...
6  | 1 2185 2587279 2698167665 2711895924889 2692324030864335 ...
   ...

Links

Crossrefs

Rows 1..4 are A000012, A107839(n-1), A158453, A359991.
Main diagonal is A359992.
Cf. A116469 (spanning trees), A287151 (connected induced subgraphs), A286912 (edge covers), A359990 (edge cuts), A360194 (spanning forests).

Formula

T(m,n) = T(n,m).

A359987 Number of edge cuts in the n-ladder graph P_2 X P_n.

Original entry on oeis.org

1, 11, 105, 919, 7713, 63351, 514321, 4148839, 33347041, 267489431, 2143168305, 17160184519, 137349160833, 1099102033911, 8794224638161, 70360221445159, 562911076526881, 4503422288363351, 36027988077717105, 288226686123491719, 2305826176955087553, 18446667292472959671
Offset: 1

Views

Author

Andrew Howroyd, Jan 28 2023

Keywords

Links

Crossrefs

Row 2 of A359990.
Cf. A013730, A107839, A356828 (vertex cuts), A359989.
Cf. A359621, A359622.

Programs

  • Mathematica
    LinearRecurrence[{13, -42, 16}, {1, 11, 105}, 25] (* Paolo Xausa, Jun 24 2024 *)
Table[2^(3 n - 2) + (((5 - Sqrt[17])/2)^n - ((5 + Sqrt[17])/2)^n)/Sqrt[17], {n, 20}] // Expand (* Eric W. Weisstein, Nov 03 2024 *)
CoefficientList[Series[-(1 - 2 x + 4 x^2)/((-1 + 8 x) (1 - 5 x + 2 x^2)), {x, 0, 20}], x] (* Eric W. Weisstein, Nov 03 2024 *)
  • PARI
    Vec((1 - 2*x + 4*x^2)/((1 - 8*x)*(1 - 5*x + 2*x^2)) + O(x^25))

Formula

a(n) = 13*a(n-1) - 42*a(n-2) + 16*a(n-3) for n > 3.
a(n) = A013730(n-1) - A107839(n-1).
G.f.: x*(1 - 2*x + 4*x^2)/((1 - 8*x)*(1 - 5*x + 2*x^2)).

A359988 Number of edge cuts in the 3 X n grid graph.

Original entry on oeis.org

3, 105, 3665, 123215, 4051679, 131630449, 4248037953, 136587740399, 4382607093471, 140457446235441, 4498520188148993, 144023056568886959, 4610014925578108703, 147543642097619999089, 4721816707356538941633, 151105755554498621737583, 4835522406931884652356447
Offset: 1

Views

Author

Andrew Howroyd, Jan 28 2023

Keywords

Links

Crossrefs

Row 3 of A359990.
Cf. A013823, A158453, A359987.

Programs

  • PARI
    Vec((3 - 57*x + 326*x^2 - 280*x^3 + 32*x^4)/((1 - 32*x)*(1 - 22*x + 73*x^2 - 54*x^3 + 8*x^4)) + O(x^20))

Formula

a(n) = 54*a(n-1) - 777*a(n-2) + 2390*a(n-3) - 1736*a(n-4) + 256*a(n-5) for n > 5.
G.f.: x*(3 - 57*x + 326*x^2 - 280*x^3 + 32*x^4)/((1 - 32*x)*(1 - 22*x + 73*x^2 - 54*x^3 + 8*x^4)).
a(n) = A013823(n-1) - A158453(n).

A359989 Number of edge cuts in the n X n grid graph.

Original entry on oeis.org

0, 11, 3665, 16222021, 1089224690733, 1150229180575982641, 19332960184149905415396841, 5191795779341747010730275139276101, 22300392508126626453613075681944644703634073, 1532492114568208344599582951444206568577948454351297761
Offset: 1

Views

Author

Andrew Howroyd, Jan 28 2023

Keywords

Examples

			The a(2) = 11 edge cuts are the following subgraphs and their rotations and reflections. The illustrations show the edge sets that must be removed to disconnect the graph.
    o---o    o---o   o---o   o---o
    |                |       |   |
    o   o    o---o   o---o   o---o
The first two of these show minimal edge cuts.

Links

  • Eric Weisstein's World of Mathematics, Edge Cut
  • Eric Weisstein's World of Mathematics, Grid Graph

Crossrefs

Main diagonal of A359990.
Cf. A053765, A286913, A359992.

Formula

a(n) = A053765(n) - A359992(n).
Showing 1-5 of 5 results.

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