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 1-3 of 3 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).

A378934 Number of minimal edge cuts in the 4 X n grid graph.

Original entry on oeis.org

3, 28, 146, 627, 2471, 9292, 33878, 120771, 423251, 1463908, 5011690, 17021179, 57450167, 192966908, 645696454, 2154226075, 7170606795, 23825657596, 79055534746, 262031761435, 867792229799, 2872103661988, 9501035284286, 31417942222787, 103862506390523, 343276150243020
Offset: 1

Views

Author

Andrew Howroyd, Dec 11 2024

Keywords

Links

Crossrefs

Row 4 of A378932.
Cf. A166766, A378933.

Formula

G.f.: x*(3 - 2*x - 23*x^2 + 26*x^3 + 32*x^4 - 2*x^5 - 29*x^6 - 18*x^7 - 3*x^8)/((1 - x)^3*(1 - 2*x - x^2)^2*(1 - 3*x - x^2)).
a(n) = 10*a(n-1) - 37*a(n-2) + 59*a(n-3) - 27*a(n-4) - 25*a(n-5) + 21*a(n-6) + 5*a(n-7) - 4*a(n-8) - a(n-9) for n >= 10.
a(n) = A166766(n)/2.

A166761 Number of n X 3 1..2 arrays containing at least one of each value, and all equal values connected.

Original entry on oeis.org

4, 30, 106, 292, 712, 1618, 3518, 7432, 15404, 31526, 63986, 129164, 259824, 521498, 1045254, 2093232, 4189716, 8383278, 16771066, 33547380, 67100824, 134208610, 268425166, 536859352, 1073728892, 2147469238, 4294951298, 8589916892
Offset: 1

Views

Author

R. H. Hardin, Oct 21 2009

Keywords

Examples

			Some solutions for n=4
...2.2.2...2.2.2...1.1.2...1.1.1...2.1.1...1.1.1...1.2.2...1.2.2...1.1.1
...2.1.1...1.2.2...1.2.2...2.2.1...2.2.2...2.1.1...1.2.2...1.2.2...1.2.1
...2.1.1...1.2.1...1.2.2...2.2.1...2.2.2...2.2.1...1.1.2...1.1.2...1.2.1
...2.2.1...1.1.1...1.1.2...2.2.1...2.2.2...2.2.2...2.2.2...1.1.1...1.1.1
------
...1.2.2...2.2.2...1.1.2...1.1.1...1.1.1...1.1.2...1.2.2...1.2.2...1.1.1
...1.1.2...2.1.2...1.1.2...2.2.1...1.1.2...1.1.2...1.1.2...1.2.1...1.1.1
...1.1.2...1.1.1...1.2.2...2.2.1...1.1.2...1.1.2...1.1.2...1.1.1...2.2.1
...1.1.1...1.1.1...1.1.1...1.1.1...1.1.1...1.1.1...2.2.2...1.1.1...2.1.1

Links

Crossrefs

Twice row 3 of A378932.
Cf. A378933.

Formula

Empirical: a(n) = 6*a(n-1) - 14*a(n-2) + 16*a(n-3) -9*a(n-4) + 2*a(n-5).
From G. C. Greubel, May 26 2016: (Start)
Empirical a(n) = (3*2^(n + 5) - 2*n^3 - 9*n^2 - 73*n - 96)/3.
Empirical G.f.: (1/3)*( 96/(1 - 2*x) + 6*(-16 + 34*x - 25*x^2 + 5*x^3)/(1 - x)^4 ).
Empirical E.g.f.: (1/3)*(96*exp(x) - (96 + 84*x + 15*x^2 + 2*x^3 ) )*exp(x). (End)
From Andrew Howroyd, Dec 12 2024: (Start)
The above empirical formulas are correct.
a(n) = 2*A378933(n).
(End)
Showing 1-3 of 3 results.

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