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.

A218354 T(n,k) = Hilltop maps: number of nXk binary arrays indicating the locations of corresponding elements not exceeded by any horizontal or vertical neighbor in a random 0..1 n X k array.

Original entry on oeis.org

1, 3, 3, 5, 11, 5, 9, 41, 41, 9, 17, 149, 291, 149, 17, 31, 547, 2069, 2069, 547, 31, 57, 2007, 14811, 28661, 14811, 2007, 57, 105, 7361, 105913, 401253, 401253, 105913, 7361, 105, 193, 27001, 757305, 5609569, 10982565, 5609569, 757305, 27001, 193, 355
Offset: 1

Views

Author

R. H. Hardin, Oct 26 2012

Keywords

Comments

From Andrew Howroyd, May 10 2017: (Start)
Number of n X k binary matrices with every 1 vertically or horizontally adjacent to some 0.
Number of dominating sets in the grid graph P_n X P_k. (End)

Examples

			Table starts
....1.......3...........5..............9.................17
....3......11..........41............149................547
....5......41.........291...........2069..............14811
....9.....149........2069..........28661.............401253
...17.....547.......14811.........401253...........10982565
...31....2007......105913........5609569..........300126903
...57....7361......757305.......78394141.........8199377227
..105...27001.....5415209.....1095695529.......224032447213
..193...99043....38722037....15314367301......6121258910011
..355..363299...276885777...214044940145....167250519310183
..653.1332617..1979899795..2991651891557...4569773233045519
.1201.4888173.14157473937.41813576818545.124859601874166153
...
Some solutions for n=3 k=4
..1..0..1..1....1..1..1..0....1..1..1..0....1..0..1..1....1..0..1..1
..1..0..1..0....1..0..1..0....0..0..1..0....1..0..1..1....1..1..0..1
..0..0..1..0....1..1..0..1....0..1..1..1....1..1..1..1....1..1..1..0

Links

Crossrefs

Columns 1-7 are A000213(n+1), A218348, A218349, A218350, A218351, A218352, A218353.
Diagonal is A133515.
Cf. A089934 (independent vertex sets), A210662 (matchings).

Formula

Empirical for column k:
k=1: a(n) = a(n-1) +a(n-2) +a(n-3).
k=2: a(n) = 3*a(n-1) +2*a(n-2) +2*a(n-3) -a(n-4) -a(n-5).
k=3: a(n) = 6*a(n-1) +5*a(n-2) +22*a(n-3) +7*a(n-4) +8*a(n-5) -18*a(n-6) -20*a(n-7) -a(n-8) +4*a(n-9) +3*a(n-10) +a(n-12).
Column k=1 for an underlying 0..z array: a(n) = sum(i=1..2z+1){a(n-i)} z=1,2,3,4

A219078 T(n,k)=Hilltop maps: number of nXk binary arrays indicating the locations of corresponding elements not exceeded by any horizontal, diagonal or antidiagonal neighbor in a random 0..1 nXk array.

Original entry on oeis.org

1, 3, 1, 5, 11, 1, 9, 47, 41, 1, 17, 165, 337, 149, 1, 31, 625, 2321, 2469, 547, 1, 57, 2435, 17537, 32945, 18499, 2007, 1, 105, 9367, 134809, 494713, 477309, 137251, 7361, 1, 193, 35901, 1023441, 7561349, 14228041, 6879341, 1019123, 27001, 1, 355, 137865
Offset: 1

Views

Author

R. H. Hardin Nov 11 2012

Keywords

Comments

Table starts
.1.......3...........5..............9.................17....................31
.1......11..........47............165................625..................2435
.1......41.........337...........2321..............17537................134809
.1.....149........2469..........32945.............494713...............7561349
.1.....547.......18499.........477309...........14228041.............433704331
.1....2007......137251........6879341..........407374825...........24734141495
.1....7361.....1019123.......99118753........11660290321.........1410242020653
.1...27001.....7573641.....1428782305.......333882416305........80444813963129
.1...99043....56263253....20594013941......9559854123673......4588436794733591
.1..363299...417979331...296830835781....273717397488937....261712432286491659
.1.1332617..3105269893..4278398369137...7837125931615553..14927540586501299193
.1.4888173.23069495037.61667023808785.224393896465841065.851435525780779197693

Examples

			Some solutions for n=3 k=4
..1..1..0..0....1..1..1..1....1..0..0..1....1..1..1..1....0..0..1..1
..0..0..1..0....1..0..0..1....1..1..0..0....1..1..1..0....1..1..1..0
..0..1..0..0....1..1..0..1....1..0..1..1....0..1..0..1....0..0..0..1

Links

Crossrefs

Column 2 is A218348
Row 1 is A000213(n+1)

A284702 Number of dominating sets in the n-prism graph.

Original entry on oeis.org

3, 11, 51, 183, 663, 2435, 8935, 32775, 120219, 440971, 1617531, 5933271, 21763823, 79831875, 292831311, 1074134535, 3940032883, 14452434635, 53012975555, 194456895863, 713287340551, 2616409296963, 9597250953527, 35203676264199, 129130605057163
Offset: 1

Views

Author

Eric W. Weisstein, Apr 01 2017

Keywords

Comments

Sequence extrapolated to n=1 using recurrence. - Andrew Howroyd, May 10 2017

Links

Crossrefs

Cf. A051927, A102080, A284699, A218348 (ladder), A284663.

Programs

  • Mathematica
    LinearRecurrence[{3, 1, 5, 1, 1, -1, -1}, {3, 11, 51, 183, 663, 2435,
  8935}, 20] (* Eric W. Weisstein, May 17 2017 *)
Rest[CoefficientList[Series[x (-7 x^6 - 6 x^5 + 5 x^4 + 4 x^3 + 15 x^2 + 2 x + 3)/((x^2 + 1) (x^5 + x^4 - 2 x^3 - 2 x^2 - 3 x + 1)), {x, 0, 20}], x]] (* G. C. Greubel, May 17 2017 *)
Table[2 Cos[n Pi/2] + RootSum[1 + #1 - 2 #1^2 - 2 #1^3 - 3 #1^4 + #1^5 &, #^n &], {n, 20}] (* Eric W. Weisstein, May 26 2017 *)
  • PARI
    Vec((-7*x^6-6*x^5+5*x^4+4*x^3+15*x^2+2*x+3)/((x^2+1)*(x^5+x^4-2*x^3-2*x^2-3*x+1))+O(x^15)) \\ Andrew Howroyd, May 10 2017

Formula

From Andrew Howroyd, May 10 2017: (Start)
a(n) = 3*a(n-1) + a(n-2) + 5*a(n-3) + a(n-4) + a(n-5) - a(n-6) - a(n-7).
G.f.: x*(-7*x^6 - 6*x^5 + 5*x^4 + 4*x^3 + 15*x^2 + 2*x + 3)/((x^2 + 1)*(x^5 + x^4 - 2*x^3 - 2*x^2 - 3*x + 1)). (End)

Extensions

a(1)-a(2) and a(16)-a(25) from Andrew Howroyd, May 10 2017

A284663 Number of dominating sets in the Moebius ladder M_n.

Original entry on oeis.org

3, 15, 51, 179, 663, 2439, 8935, 32771, 120219, 440975, 1617531, 5933267, 21763823, 79831879, 292831311, 1074134531, 3940032883, 14452434639, 53012975555, 194456895859, 713287340551, 2616409296967, 9597250953527, 35203676264195, 129130605057163
Offset: 1

Views

Author

Eric W. Weisstein, Mar 31 2017

Keywords

Comments

Sequence extrapolated to n=1 using recurrence. - Andrew Howroyd, May 10 2017

Links

Crossrefs

Cf. A182143, A284702, A218348 (ladder).

Programs

  • Mathematica
    LinearRecurrence[{3, 1, 5, 1, 1, -1, -1}, {3, 15, 51, 179, 663, 2439,
  8935}, 20] (* Eric W. Weisstein, May 17 2017 *)
Rest[CoefficientList[Series[x*(1 - x)*(1 + x)*(3*x^4 + 2*x^3 + 6*x^2 + 6*x + 3)/((x^2 + 1)*(x^5 + x^4 - 2*x^3 - 2*x^2 - 3*x + 1)), {x,0,50}], x]] (* G. C. Greubel, May 17 2017 *)
Table[RootSum[1 + # - 2 #^2 - 2 #^3 - 3 #^4 + #^5 &, #^n &] - 2 Cos[n Pi/2], {n, 20}] (* Eric W. Weisstein, Jun 14 2017 *)
  • PARI
    Vec((1-x)*(1+x)*(3*x^4+2*x^3+6*x^2+6*x+3)/((x^2+1)*(x^5+x^4-2*x^3-2*x^2-3*x+1))+O(x^50)) \\ Andrew Howroyd, May 10 2017

Formula

From Andrew Howroyd, May 10 2017 (Start)
a(n) = 3*a(n-1)+a(n-2)+5*a(n-3)+a(n-4)+a(n-5)-a(n-6)-a(n-7) for n>7.
G.f.: x*(1-x)*(1+x)*(3*x^4+2*x^3+6*x^2+6*x+3)/((x^2+1)*(x^5+x^4-2*x^3 -2*x^2-3*x+1)). (End)

Extensions

a(1)-(2) and a(16)-a(25) from Andrew Howroyd, May 10 2017
Showing 1-4 of 4 results.

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