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.

Previous Showing 11-15 of 15 results.

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

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

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

A362574 Number of vertex cuts in the n X n queen graph.

Original entry on oeis.org

0, 0, 16, 720, 76268, 24883487
Offset: 1

Views

Author

Eric W. Weisstein, Apr 25 2023

Keywords

Links

Crossrefs

Cf. A285765 (number of connected induced subgraphs).
Previous Showing 11-15 of 15 results.

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