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.

A286847 Array read by antidiagonals: T(m,n) = number of minimal dominating sets in the grid graph P_m X P_n.

Original entry on oeis.org

1, 2, 2, 2, 6, 2, 4, 7, 7, 4, 4, 18, 16, 18, 4, 7, 39, 53, 53, 39, 7, 9, 75, 154, 306, 154, 75, 9, 13, 155, 436, 1167, 1167, 436, 155, 13, 18, 310, 1268, 4939, 6958, 4939, 1268, 310, 18, 25, 638, 3660, 21313, 40931, 40931, 21313, 3660, 638, 25
Offset: 1

Views

Author

Andrew Howroyd, Aug 01 2017

Keywords

Examples

			Table begins:
===============================================================
m\n|  1   2    3     4       5        6         7          8
---|-----------------------------------------------------------
1  |  1   2    2     4       4        7         9         13...
2  |  2   6    7    18      39       75       155        310...
3  |  2   7   16    53     154      436      1268       3660...
4  |  4  18   53   306    1167     4939     21313      88161...
5  |  4  39  154  1167    6958    40931    254754    1519544...
6  |  7  75  436  4939   40931   349178   3118754   26797630...
7  |  9 155 1268 21313  254754  3118754  40307167  497709474...
8  | 13 310 3660 88161 1519544 26797630 497709474 8863408138...
...

Links

Crossrefs

Rows 1-3 are A253412, A290379, A286848.
Main diagonal is A290382.
Cf. A218354 (dominating sets), A089934 (independent), A286868 (irredundant).
Cf. A286849 (king graph).

A290336 Number of minimal dominating sets in the n-prism graph.

Original entry on oeis.org

11, 12, 37, 55, 149, 316, 596, 1219, 2444, 4971, 10103, 20465, 41746, 84924, 172501, 350668, 712597, 1448447, 2943959, 5983344, 12162310, 24720787, 50246512, 102129655, 207584129, 421928981, 857596064, 1743117100, 3543000201, 7201373724, 14637255611
Offset: 3

Views

Author

Eric W. Weisstein, Jul 27 2017

Keywords

Comments

The prism graphs are defined for n>=3. If the sequence is extended to n=1 using P_n X P_2 then a(1)=2 and a(2)=6 (as A290379). The empirical recurrence is the same as that for the Moebius ladder graph (see A290337). - Andrew Howroyd, Aug 01 2017

Links

Crossrefs

Cf. A284702, A290337, A290379.

Formula

Empirical: a(n) = a(n-1)+a(n-2)+2*a(n-3) -a(n-4)+2*a(n-5)-2*a(n-6) +6*a(n-7)+4*a(n-8)+4*a(n-9) -6*a(n-10)-3*a(n-12) +5*a(n-13)-a(n-14)-2*a(n-15) -5*a(n-16)-2*a(n-17)-2*a(n-18) for n > 20. - Andrew Howroyd, Aug 01 2017
Empirical g.f.: x^3*(11 + x + 14*x^2 - 16*x^3 + 44*x^4 + 28*x^5 + 56*x^6 - 52*x^7 - 6*x^8 - 70*x^9 + 52*x^10 - 28*x^11 - 23*x^12 - 97*x^13 - 56*x^14 - 62*x^15 - 12*x^16 - 8*x^17) / ((1 - x)*(1 + 2*x^4 + x^6)*(1 - x^2 - 3*x^3 - 4*x^4 - 4*x^5 - x^6 - 2*x^7 - 3*x^8 - 5*x^9 - 4*x^10 - 2*x^11)). - Colin Barker, Aug 02 2017

Extensions

Terms a(9) and beyond from Andrew Howroyd, Aug 01 2017

A290337 Number of minimal dominating sets in the n-Moebius ladder.

Original entry on oeis.org

2, 4, 11, 28, 37, 67, 149, 284, 596, 1179, 2444, 5023, 10103, 20577, 41746, 84860, 172501, 350392, 712597, 1448463, 2943959, 5983960, 12162310, 24721031, 50246512, 102128407, 207584129, 421927877, 857596064, 1743119352, 3543000201, 7201377180, 14637255611
Offset: 1

Views

Author

Eric W. Weisstein, Jul 27 2017

Keywords

Links

Crossrefs

Cf. A284663, A290336, A290379.

Formula

Empirical: a(n) = a(n-1)+a(n-2)+2*a(n-3) -a(n-4)+2*a(n-5)-2*a(n-6) +6*a(n-7)+4*a(n-8)+4*a(n-9) -6*a(n-10)-3*a(n-12) +5*a(n-13)-a(n-14)-2*a(n-15) -5*a(n-16)-2*a(n-17)-2*a(n-18) for n > 18. - Andrew Howroyd, Aug 01 2017
Empirical g.f.: x*(2 + 2*x + 5*x^2 + 9*x^3 - 8*x^4 - 20*x^5 - 4*x^6 - 4*x^7 - 40*x^9 - 26*x^10 - 26*x^11 + 14*x^12 - 22*x^13 - 33*x^14 - 45*x^15 - 14*x^16 - 14*x^17) / ((1 - x)*(1 + 2*x^4 + x^6)*(1 - x^2 - 3*x^3 - 4*x^4 - 4*x^5 - x^6 - 2*x^7 - 3*x^8 - 5*x^9 - 4*x^10 - 2*x^11)). - Colin Barker, Aug 02 2017

Extensions

a(1)-a(2) and terms a(9) and beyond from Andrew Howroyd, Aug 01 2017

A303072 Number of minimal total dominating sets in the n-ladder graph.

Original entry on oeis.org

1, 4, 4, 16, 16, 49, 81, 169, 324, 625, 1296, 2401, 4900, 9409, 18769, 36481, 71824, 141376, 276676, 544644, 1067089, 2099601, 4116841, 8088336, 15880225, 31181056, 61230625, 120209296, 236083225, 463497841, 910168561, 1787091076, 3509140644, 6890328064, 13529411856
Offset: 1

Views

Author

Eric W. Weisstein, Apr 18 2018

Keywords

Links

Crossrefs

Row 2 of A303118.
Cf. A290379, A302402, A303054.

Programs

  • Mathematica
    Table[(RootSum[1 - #^2 - #^3 - #^4 + #^6 &, (9 - 18 #^2 + 23 #^3 - 3 #^4 + 32 #^5) #^n &]/229)^2, {n, 40}]
LinearRecurrence[{-1, 1, 3, 7, 8, 2, 6, 6, 0, 0, -6, -6, -2, -8, -7, -3, -1, 1, 1}, {1, 4, 4, 16, 16, 49, 81, 169,324, 625, 1296, 2401, 4900, 9409, 18769, 36481, 71824, 141376, 276676}, 40]
CoefficientList[Series[(-1 - 5 x - 7 x^2 - 13 x^3 - 9 x^4 - x^5 - 4 x^6 + 5 x^7 + 13 x^8 + 14 x^9 + 21 x^10 + 15 x^11 + 12 x^12 + 15 x^13 + 9 x^14 + 3 x^15 - 2 x^17 - x^18)/(-1 - x + x^2 + 3 x^3 + 7 x^4 + 8 x^5 + 2 x^6 + 6 x^7 + 6 x^8 - 6 x^11 - 6 x^12 - 2 x^13 - 8 x^14 - 7 x^15 - 3 x^16 - x^17 + x^18 + x^19), {x, 0, 40}], x]

Formula

a(n) = A253412(n)^2.
G.f.: x*(-1 - 5*x - 7*x^2 - 13*x^3 - 9*x^4 - x^5 - 4*x^6 + 5*x^7 + 13*x^8 + 14*x^9 + 21*x^10 + 15*x^11 + 12*x^12 + 15*x^13 + 9*x^14 + 3*x^15 - 2*x^17 - x^18)/(-1 - x + x^2 + 3*x^3 + 7*x^4 + 8*x^5 + 2*x^6 + 6*x^7 + 6*x^8 - 6*x^11 - 6*x^12 - 2*x^13 - 8*x^14 - 7*x^15 - 3*x^16 - x^17 + x^18 + x^19).
Showing 1-4 of 4 results.

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

Content is available under The OEIS End-User License Agreement.