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.

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A285765 Number of connected induced (non-null) subgraphs of the n X n queen graph.

Original entry on oeis.org

1, 15, 495, 64815, 33478163, 68694593248
Offset: 1

Views

Author

Giovanni Resta, May 04 2017

Keywords

Links

Crossrefs

Cf. A020873 (wheel), A059020 (ladder), A059525 (grid), A286139 (king), A286182 (prism), A286183 (antiprism), A286184 (helm), A286185 (Möbius ladder), A286186 (friendship), A286187 (web), A286188 (gear), A286189 (rook).

Programs

  • Mathematica
    Table[g = GraphData[{"Queen", {n, n}}]; -1 + ParallelSum[ Boole@ ConnectedGraphQ@ Subgraph[g, s], {s, Subsets@ Range[n^2]}], {n, 4}]

A286186 Number of connected induced (non-null) subgraphs of the friendship graph with 2n+1 nodes.

Original entry on oeis.org

7, 22, 73, 268, 1039, 4114, 16405, 65560, 262171, 1048606, 4194337, 16777252, 67108903, 268435498, 1073741869, 4294967344, 17179869235, 68719476790, 274877907001, 1099511627836, 4398046511167, 17592186044482, 70368744177733, 281474976710728, 1125899906842699
Offset: 1

Views

Author

Giovanni Resta, May 04 2017

Keywords

Links

Crossrefs

Cf. A020873 (wheel), A059020 (ladder), A059525 (grid), A286139 (king), A286182 (prism), A286183 (antiprism), A286184 (helm), A286185 (Möbius ladder), A286187 (web), A286188 (gear), A286189 (rook), A285765 (queen).

Programs

  • Mathematica
    Table[4^n + 3 n, {n, 30}]
LinearRecurrence[{6,-9,4},{7,22,73},40] (* Harvey P. Dale, May 25 2019 *)
  • PARI
    Vec(x*(7 - 20*x + 4*x^2) / ((1 - x)^2*(1 - 4*x)) + O(x^30)) \\ Colin Barker, May 21 2017

Formula

a(n) = 4^n + 3*n.
From Colin Barker, May 21 2017: (Start)
G.f.: x*(7 - 20*x + 4*x^2) / ((1 - x)^2*(1 - 4*x)).
a(n) = 6*a(n-1) - 9*a(n-2) + 4*a(n-3) for n>3. (End)
E.g.f.: exp(x)*(exp(3*x) + 3*x) - 1. - Stefano Spezia, Aug 25 2022

A287690 Number of connected dominating sets in the n X n grid graph.

Original entry on oeis.org

1, 9, 129, 5617, 964755, 617429805, 1436456861467, 12128014243816259, 370157141019558632729, 40729998558184127557326187, 16129157077874837008807129310501, 22956060013827748812137293758719842059, 117308080543566432787532732819884994609487361
Offset: 1

Views

Author

Eric W. Weisstein, May 29 2017

Keywords

Links

Crossrefs

Main diagonal of A291872.
Cf. A059525, A133515, A289180.

Extensions

a(6)-a(13) from Andrew Howroyd, Sep 04 2017

A331986 Number of snake-like polyominoes with the maximum possible number of unit squares in an n X n square.

Original entry on oeis.org

1, 4, 8, 84, 56, 136, 52, 216, 16, 1504, 2352, 1152, 1344, 123216, 82432, 11008, 308992
Offset: 1

Views

Author

Alain Goupil, Feb 03 2020

Keywords

Comments

The maximum possible number of unit squares is given by A331968(n).
Equivalently, a(n) is the number of maximum length paths without chords in the n X n grid graph. A path without chords is an induced subgraph that is a path.
For n > 1, a(n) is a multiple of 4 since a solution can have at most one symmetry considering rotations and reflections. - Andrew Howroyd, Feb 04 2020

Examples

			For n = 4 the number of snake-like polyominoes with 11 cells is 84.

Links

  • Eric Weisstein's World of Mathematics, Grid Graph

Crossrefs

Main diagonal of A360916.
Cf. A331968, A059525 (connected induced subgraphs), A099155.
Cf. A332920 (non-isomorphic snakes), A332921 (symmetric snakes).

Extensions

a(15) from Andrew Howroyd, Feb 04 2020
a(16)-a(17) from Yi Yang, Oct 03 2022

A285934 Number of connected induced (non-null) subgraphs of the perfect binary tree of height n.

Original entry on oeis.org

1, 6, 37, 750, 459829, 210067308558, 44127887746326310604917, 1947270476915296449559791701269341583074001038
Offset: 0

Views

Author

Giovanni Resta, May 05 2017

Keywords

Comments

A perfect (sometimes called complete) binary tree of height k has 2^(k+1)-1 nodes.
a(8) has 91 digits and thus it is not reported.

Links

Crossrefs

Cf. A003095, A020873 (wheel), A059020 (ladder), A059525 (grid), A286139 (king), A286182 (prism), A286183 (antiprism), A286184 (helm), A286185 (Möbius ladder), A286186 (friendship), A286187 (web), A286188 (gear), A286189 (rook), A285765 (queen).

Programs

  • Mathematica
    a[1]=b[1]=1; b[n_] := b[n] = 1 + b[n - 1]^2; a[n_] := a[n] = b[n]^2 + 2 a[n - 1]; Array[a, 8]

Formula

Let b(0)=1 and b(n) = 1+b(n-1)^2. Then, a(0)=1 and a(n) = b(n)^2 + 2*a(n-1). Note that b(n) = A003095(n+1).

A286191 a(n) = (2^n-1)^2 + 2*n.

Original entry on oeis.org

3, 13, 55, 233, 971, 3981, 16143, 65041, 261139, 1046549, 4190231, 16769049, 67092507, 268402717, 1073676319, 4294836257, 17179607075, 68718952485, 274876858407, 1099509530665, 4398042316843, 17592177655853, 70368727400495, 281474943156273, 1125899839733811
Offset: 1

Views

Author

Giovanni Resta, May 05 2017

Keywords

Comments

Number of connected induced (non-null) subgraphs of the complete bipartite graph K(n,n).

Links

Crossrefs

Cf. A020873 (wheel), A059020 (ladder), A059525 (grid), A286139 (king), A286182 (prism), A286183 (antiprism), A286184 (helm), A286185 (Möbius ladder), A286186 (friendship), A286187 (web), A286188 (gear), A286189 (rook), A285765 (queen).

Programs

  • Mathematica
    a[n_] := (2^n-1)^2 + 2*n; Array[a, 30]
Table[(2^n - 1)^2 + 2 n, {n, 20}] (* Eric W. Weisstein, Aug 09 2017 *)
LinearRecurrence[{8, -21, 22, -8}, {3, 13, 55, 233}, 20] (* Eric W. Weisstein, Aug 09 2017 *)
CoefficientList[Series[(3 - 11 x + 14 x^2)/((-1 + x)^2 (1 - 6 x + 8 x^2)), {x, 0, 20}], x] (* Eric W. Weisstein, Aug 09 2017 *)
  • PARI
    Vec(x*(3 - 11*x + 14*x^2) / ((1 - x)^2*(1 - 2*x)*(1 - 4*x)) + O(x^30)) \\ Colin Barker, May 30 2017

Formula

a(n) = (2^n-1)^2 + 2*n.
From Colin Barker, May 30 2017: (Start)
G.f.: x*(3 - 11*x + 14*x^2) / ((1 - x)^2*(1 - 2*x)*(1 - 4*x)).
a(n) = 8*a(n-1) - 21*a(n-2) + 22*a(n-3) - 8*a(n-4) for n>4.
(End)

Extensions

Name changed to the formula by Eric W. Weisstein, Aug 09 2017

A360200 Number of induced paths in the n X n grid graph.

Original entry on oeis.org

0, 8, 94, 1004, 14864, 334536, 11546874, 629381852, 56094263348, 8343512638896, 2074276200162230, 853966325494701152, 578432462293854136504, 646135466408339553958096, 1200595044818176185884236342
Offset: 1

Views

Author

Andrew Howroyd, Jan 29 2023

Keywords

Comments

Paths of length zero are not counted here.
Equivalently, a(n) is the number of snake-like polyominoes in an n X n square. Rotations, reflections and translations are counted separately.

Examples

			The a(2) = 8 induced paths are:
  O O   O .   . .   . O   O O   O .   . O   O O
  . .   O .   O O   . O   O .   O O   O O   . O

Links

Crossrefs

Main diagonal of A360199.
Cf. A059525, A297664 (induced cycles), A331968, A331986 (of maximum length), A357516.

A360203 Number of (non-null) induced trees in the n X n grid graph.

Original entry on oeis.org

1, 12, 138, 3568, 277606, 66136452, 48136454388, 106601739449932, 716581962133166734, 14594259085593605592840, 899530518959027898354960664, 167638624754374503965030664785872, 94397539071875018677962029008899452442, 160524233982090828046095750880433748533447560
Offset: 1

Views

Author

Andrew Howroyd, Feb 22 2023

Keywords

Links

Crossrefs

Main diagonal of A360202.
Cf. A059525, A080691, A297664, A360200.

A286304 Number of connected induced (non-null) subgraphs of the complete binary tree with n nodes.

Original entry on oeis.org

1, 3, 6, 10, 17, 24, 37, 51, 78, 110, 173, 229, 340, 477, 750, 1024, 1571, 2253, 3616, 5024, 7839, 11356, 18389, 25173, 38740, 55697, 89610, 124870, 195389, 283536, 459829, 636123, 988710, 1429442, 2310905, 3227617, 5061040, 7352817, 11936370, 16526444
Offset: 1

Views

Author

Giovanni Resta, May 05 2017

Keywords

Links

Crossrefs

Cf. A285934, A020873 (wheel), A059020 (ladder), A059525 (grid), A286139 (king), A286182 (prism), A286183 (antiprism), A286184 (helm), A286185 (Möbius ladder), A286186 (friendship), A286187 (web), A286188 (gear), A286189 (rook), A285765 (queen).

Programs

  • Mathematica
    Join[{1}, Table[g=KaryTree[n]; -1 + ParallelSum[Boole@ConnectedGraphQ@Subgraph[g, s], {s, Subsets@Range[n]}], {n, 2, 16}]]
(* Second program: *)
l[n_] := With[{h = 2^Floor[Log[2, n]]}, Min[h - 1, n - h/2]];
b[n_] := b[n] = 1 + If[n <= 1, n, b[l[n]]*b[n - 1 - l[n]]];
a[n_] := a[n] = If[n <= 1, n, b[n] - 1 + a[l[n]] + a[n - 1 - l[n]]];
Array[a, 40] (* Jean-François Alcover, Nov 01 2017, after Andrew Howroyd *)
  • PARI
    l(n)={my(h=2^floor(log(n)/log(2))); min(h-1,n-h/2)}
b(n)=1+if(n<=1,n,b(l(n))*b(n-1-l(n)));
a(n)=if(n<=1,n,b(n)-1 + a(l(n)) + a(n-1-l(n))); \\ Andrew Howroyd, May 22 2017

Formula

a(2^k-1) = A285934(k-1).

Extensions

Terms a(35) and beyond from Andrew Howroyd, May 22 2017

A272435 Number of n-ominoes in n X n grid (i.e., rookwise connected sets of n cells in a square array with n rows and n columns).

Original entry on oeis.org

1, 4, 22, 113, 571, 2816, 13616, 64678, 302574, 1397318, 6382660, 28882214, 129640058, 577812724, 2559491834, 11276000877, 49437494408, 215815377168
Offset: 1

Views

Author

Don Knuth, Apr 29 2016

Keywords

Comments

Higher corrected values are supported by exhibiting a(n) distinct n-ominoes in the n-square for n=10 and n=11 (see LINKS below). - James Stein, Dec 11 2017
a(n) is the number of connected induced subgraphs with n vertices in the n X n grid graph. - Andrew Howroyd, Apr 27 2020

Examples

			The 22 arrangements for n=3 include three horizontal rows, three vertical rows, and four ways to place each rotation of the L-tromino.

References

  • This sequence will some day be mentioned in an exercise in section 7.2.2 of The Art of Computer Programming.

Links

Crossrefs

Cf. A001168, A059525, A162676.

Extensions

a(10)-a(12) corrected, and a(13)-a(14) added by James Stein, Dec 11 2017
a(15)-a(16) from Andrew Howroyd, Apr 27 2020
a(17)-a(18) from Giovanni Resta, May 01 2020
Previous Showing 11-20 of 22 results. Next

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