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 1-4 of 4 results.

A320097 Number of no-leaf subgraphs of the 4 X n grid.

Original entry on oeis.org

1, 15, 463, 16372, 583199, 20788249, 741026781, 26415034787, 941604528692, 33564941612743, 1196473967526971, 42650154782713601, 1520330364358307239, 54194514148101568538, 1931846809485041315873, 68863650758427752078777, 2454750745501814744040599
Offset: 1

Views

Author

Peter Kagey, Oct 05 2018

Keywords

Comments

Also, the number of ways to lay unit-length matchsticks on a 4 X n grid of points in such a way that no end is "orphaned".

Examples

			Three of the a(3) = 463 subgraphs of the 4 X 3 grid with no leaf vertices are
  +   +---+          +   +   +          +   +---+
      |   |                                 |   |
  +---+---+          +---+---+          +   +---+
  |   |    ,         |   |   |,   and            .
  +---+   +          +   +---+          +---+   +
  |   |              |   |              |   |
  +---+   +          +---+   +          +---+   +

Links

Crossrefs

A093129 is analogous for 2 X (n+1) grids.
A301976 is analogous for 3 X n grids.

Formula

Conjecture: a(n) = 36*a(n-1) - 7*a(n-2) - 201*a(n-3) + 49*a(n-4) + 20*a(n-5) - 5*a(n-6) for all n > 6.
Empirical g.f.: x*(1 - 21*x - 70*x^2 + 10*x^3 + 14*x^4 - 3*x^5) / (1 - 36*x + 7*x^2 + 201*x^3 - 49*x^4 - 20*x^5 + 5*x^6). - Colin Barker, Oct 20 2018

A320099 Number of no-leaf subgraphs of the 5 X n grid.

Original entry on oeis.org

1, 50, 5193, 583199, 65485654, 7354266811, 825905301851, 92751581627976, 10416273692997679, 1169777980482365913, 131369486228240893660, 14753177269494392259423, 1656824927874469183283433, 186066281959642930757881316, 20895787297635543757965741097
Offset: 1

Views

Author

Peter Kagey, Oct 05 2018

Keywords

Comments

Also, the number of ways to lay unit-length matchsticks on a 5 X n grid of points in such a way that no end is "orphaned".

Examples

			Three of the a(3) = 5193 subgraphs of the 5 X 3 grid with no leaf vertices are:
+---+---+      +   +   +      +   +---+
|   |   |                         |   |
+---+---+      +---+---+      +   +---+
|   |   |,     |   |   |, and          .
+---+---+      +   +---+      +---+   +
|   |   |      |   |          |   |
+---+---+      +---+   +      +---+---+
|   |   |                         |   |
+---+---+      +   +   +      +   +---+

Links

Crossrefs

A093129 is analogous for 2 X (n+1) grids.
A301976 is analogous for 3 X n grids.
A320097 is analogous for 4 X n grids.

Formula

Conjecture: a(n) = 103*a(n-1) + 1063*a(n-2) - 1873*a(n-3) - 20274*a(n-4) + 44071*a(n-5) - 10365*a(n-6) - 20208*a(n-7) + 5959*a(n-8) + 2300*a(n-9) - 500*a(n-10) for n > 10.

A303930 Number of no-leaf subgraphs of the 2 X n grid up to horizontal and vertical reflection.

Original entry on oeis.org

1, 2, 4, 10, 26, 76, 232, 750, 2493, 8514, 29524, 103708, 367225, 1308542, 4682276, 16807286, 60462082, 217855460, 785863048, 2837177434, 10249053629, 37039804078, 133902392980, 484178868612, 1751030978481, 6333341963706, 22909148647012, 82872738727330
Offset: 1

Views

Author

Peter Kagey, May 02 2018

Keywords

Comments

The limit lim_{n -> infinity} A020876(n - 1)/a(n) = 4.

Examples

			For n = 4 the a(4) = 10 subgraphs of the 2 X 4 grid are:
+   +   +   +  +---+   +   +  +   +---+   +
               |   |              |   |
+   +   +   +, +---+   +   +, +   +---+   +,
+---+   +---+  +---+---+   +  +---+---+---+
|   |   |   |  |       |      |       |   |
+---+   +---+, +---+---+   +, +---+---+---+,
+---+---+---+  +---+---+---+  +---+---+---+
|           |  |   |   |   |  |   |   |   |
+---+---+---+, +---+---+---+, +---+   +---+, and
+---+---+   +
|   |   |
+---+---+   +.

Links

Crossrefs

Cf. A020876, A301976.
A093129 is analogous for 2 X (n+1) grids where reflections are considered distinct.

Formula

Conjectures from Colin Barker, May 03 2018: (Start)
G.f.: x*(1 - 6*x + 4*x^2 + 30*x^3 - 45*x^4 - 22*x^5 + 60*x^6 - 20*x^7) / ((1 - 3*x + x^2)*(1 - 5*x + 5*x^2)*(1 - 5*x^2 + 5*x^4)).
a(n) = 8*a(n-1) - 16*a(n-2) - 20*a(n-3) + 95*a(n-4) - 60*a(n-5) - 80*a(n-6) + 100*a(n-7) - 25*a(n-8) for n>8.
(End)

A320101 Table read by rows: T(n,k) is the number of no-leaf subgraphs of the n X k grid where 1 <= k <= n.

Original entry on oeis.org

1, 1, 2, 1, 5, 43, 1, 15, 463, 16372, 1, 50, 5193, 583199, 65485654, 1, 175, 58653, 20788249, 7354266811, 2602065897364, 1, 625, 663203, 741026781, 825905301851
Offset: 1

Views

Author

Peter Kagey, Oct 05 2018

Keywords

Examples

			Three of the T(4,3) = 463 subgraphs of the 4 X 3 grid with no leaf vertices are
  +   +---+      +   +   +      +   +---+
      |   |                         |   |
  +---+---+      +---+---+      +   +---+
  |   |    ,     |   |   |, and          .
  +---+   +      +   +---+      +---+   +
  |   |          |   |          |   |
  +---+   +      +---+   +      +---+   +
Table begins:
  n\k|    1    2     3        4          5             6
  ---+---------------------------------------------------
   1 |    1
   2 |    1    2
   3 |    1    5    43
   4 |    1   15   463    16372
   5 |    1   50  5193   583199   65485654
   6 |    1  175 58653 20788249 7354266811 2602065897364

Crossrefs

Cf. A093129, A301976, A320097, A320099.
Showing 1-4 of 4 results.

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