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.

A303293 Array read by antidiagonals: T(m,n) = number of minimum total dominating sets in the grid graph P_m X P_n.

Original entry on oeis.org

0, 1, 1, 2, 4, 2, 1, 1, 1, 1, 1, 16, 2, 16, 1, 4, 9, 1, 1, 9, 4, 3, 1, 3, 16, 3, 1, 3, 1, 64, 4, 256, 256, 4, 64, 1, 2, 16, 4, 4, 160, 4, 4, 16, 2, 9, 1, 9, 121, 25, 25, 121, 9, 1, 9, 4, 169, 12, 2916, 268, 144, 268, 2916, 12, 169, 4
Offset: 1

Views

Author

Andrew Howroyd, Apr 20 2018

Keywords

Comments

The minimum size of a total dominating set is the total domination number A300358(m, n).

Examples

			Table begins:
===============================================
m\n| 1  2  3    4    5     6   7    8     9
---+-------------------------------------------
1  | 0  1  2    1    1     4   3    1     2 ...
2  | 1  4  1   16    9     1  64   16     1 ...
3  | 2  1  2    1    3     4   4    9    12 ...
4  | 1 16  1   16  256     4 121 2916    25 ...
5  | 1  9  3  256  160    25 268 4225   510 ...
6  | 4  1  4    4   25   144 529 2025 10404 ...
7  | 3 64  4  121  268   529   4  441   630 ...
8  | 1 16  9 2916 4225  2025 441  256     9 ...
9  | 2  1 12   25  510 10404 630    9  1364 ...
...

Links

Crossrefs

Rows 1..2 are A302654, A303054.
Main diagonal is A303142.
Cf. A300358, A303111, A303118.

A302488 Total domination number of the n X n grid graph.

Original entry on oeis.org

0, 1, 2, 3, 6, 9, 12, 15, 20, 25, 30, 35, 42, 49, 56, 63, 72, 81, 90, 99, 110, 121, 132, 143, 156, 169, 182, 195, 210, 225, 240, 255, 272, 289, 306, 323, 342, 361, 380, 399, 420, 441, 462, 483, 506, 529, 552, 575, 600, 625, 650, 675, 702, 729, 756, 783, 812, 841, 870, 899, 930
Offset: 0

Views

Author

Eric W. Weisstein, Apr 08 2018

Keywords

Comments

Extended to a(0) and a(1) using the formula/recurrence. The total domination number of the 1 X 1 grid graph is undefined.

Links

Crossrefs

Main diagonal of A300358.
The four quadrasections are A002943, A016754, A002939(n+1), A000466(n+1).
Bisections are A002378 and A085046.
Cf. A303142.

Programs

  • Magma
    R:=RealField(); [Round(((-1)^n + 2*n*(n + 2) + 4*Sin(n*Pi(R)/2) - 1)/8): n in [0..30]]; // G. C. Greubel, Apr 09 2018
  • Mathematica
    Table[(-1 + (-1)^n + 2 n (2 + n) + 4 Sin[n Pi/2])/8, {n, 0, 20}]
LinearRecurrence[{2, -1, 0, 1, -2, 1}, {0, 1, 2, 3, 6, 9}, 20]
CoefficientList[Series[x (-1 - 2 x^3 + x^4)/((-1 + x)^3 (1 + x + x^2 + x^3)), {x, 0, 20}], x]
  • PARI
    for(n=0,30, print1(round(((-1)^n + 2*n*(n + 2) + 4*sin(n*Pi/2) - 1)/8), ", ")) \\ G. C. Greubel, Apr 09 2018
  • PARI
    a(n)=my(m=n\4); (2*m+1)*(2*m + n%4) \\ Andrew Howroyd, Aug 17 2025

Formula

a(n) = ((-1)^n + 2*n*(n + 2) + 4*sin(n*Pi/2) - 1)/8.
a(n) = 2*a(n-1) - a(n-2) + a(n-4) - 2*a(n-5) + a(n-6).
G.f.: x*(1 + 2*x^3 - x^4)/((1 - x)^3*(1 + x + x^2 + x^3)).
a(4*m + r) = (2*m + 1)*(2*m + r) for 0 <= r < 4. - Charles Kusniec, Aug 16 2025
From Amiram Eldar, Aug 26 2025: (Start)
Sum_{n>=1} 1/a(n) = Pi^2/8 + 3/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/8 - 1/2. (End)

Extensions

a(0)=0 prepended and offset corrected by Andrew Howroyd, Aug 17 2025

A303161 Number of minimal total dominating sets in the n X n grid graph.

Original entry on oeis.org

0, 4, 6, 169, 2622, 137641, 11458758, 1944369025, 692690245830, 490393052832400, 695395811259688914, 1963720302048546357904, 11300709997961358290597645, 129580789221471473285725965124, 2990581397819168926985646623641461
Offset: 1

Views

Author

Eric W. Weisstein, Apr 19 2018

Keywords

Links

Crossrefs

Main diagonal of A303118.
Cf. A133793, A290382, A303142.

Formula

a(n) = A303118(n,n).

Extensions

a(7)-a(15) from Andrew Howroyd, Apr 20 2018
Showing 1-3 of 3 results.

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