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 1-10 of 16 results. Next

A059525 Number of nonzero n X n binary arrays with all 1's connected.

Original entry on oeis.org

0, 1, 13, 218, 11506, 2301877, 1732082741, 4872949974666, 51016818604894742, 1980555831431088025753, 284374318545830329487309785, 150730745416633777472365437495914, 294516896499779486414143877573183893666, 2119097214294718323017954923662829194285541981
Offset: 0

Views

Author

David Radcliffe, Jan 21 2001

Keywords

Comments

Old name was: "Number of n X n checkerboards in which the set of red squares is edge connected".
Also the number of connected induced (non-null) subgraphs of the n X n grid graph P_n x P_n. - Eric W. Weisstein, May 01 2017

Links

Crossrefs

Main diagonal of A287151.
Cf. A059021, A020873 (wheel), A059020 (ladder), A286139 (king), A286182 (prism), A286183 (antiprism), A286184 (helm), A286185 (Möbius ladder), A286186 (friendship), A286187 (web), A286188 (gear), A286189 (rook), A285765 (queen).

Programs

Extensions

One more term from John W. Layman, Jan 25 2001
More terms from R. H. Hardin, Feb 28 2002
Clearer name from R. H. Hardin, Jul 06 2009
a(8)-a(9) from Giovanni Resta, May 03 2017
a(10)-a(13) from Andrew Howroyd, May 20 2017

A286189 Number of connected induced (non-null) subgraphs of the n X n rook graph.

Original entry on oeis.org

1, 13, 397, 55933, 31450861, 67253507293, 559182556492477, 18408476382988290493, 2416307646576708948065581, 1267404418454077249779938768413, 2658301080374793666228695738368407037, 22300360304310794054520197736231374212892413
Offset: 1

Views

Author

Giovanni Resta, May 04 2017

Keywords

Links

Crossrefs

Main diagonal of A360873.
Cf. A262307, A183109.
Cf. A020873 (wheel), A059020 (ladder), A059525 (grid), A286139 (king), A286182 (prism), A286183 (antiprism), A286184 (helm), A286185 (Möbius ladder), A286186 (friendship), A286187 (web), A286188 (gear), A285765 (queen).

Programs

  • Mathematica
    {1} ~ Join ~ Table[g = GraphData[{"Rook", {n,n}}]; -1 + ParallelSum[ Boole@ ConnectedGraphQ@ Subgraph[g, s], {s, Subsets@ Range[n^2]}], {n, 2, 4}]
(* Second program: *)
(* b = A183109, T = A262307 *)
b[n_, m_] := Sum[(-1)^j*Binomial[m, j]*(2^(m - j) - 1)^n, {j, 0, m}];
T[m_, n_] := T[m, n] = b[m, n] - Sum[T[i, j]*b[m - i, n - j] Binomial[m - 1, i - 1]*Binomial[n, j], {i, 1, m - 1}, {j, 1, n - 1}];
a[n_] := Sum[Binomial[n, i]*Binomial[n, j]*T[i, j], {i, 1, n}, {j, 1, n}];
Array[a, 12] (* Jean-François Alcover, Oct 11 2017, after Andrew Howroyd *)
  • PARI
    G(N)={my(S=matrix(N,N), T=matrix(N,N), U=matrix(N,N));
\\ S is A183109, T is A262307, U is mxn variant of this sequence.
for(m=1,N,for(n=1,N,
S[m,n]=sum(j=0, m, (-1)^j*binomial(m, j)*(2^(m - j) - 1)^n);
T[m,n]=S[m,n]-sum(i=1, m-1, sum(j=1, n-1, T[i,j]*S[m-i,n-j]*binomial(m-1,i-1)*binomial(n,j)));
U[m,n]=sum(i=1,m,sum(j=1,n,binomial(m,i)*binomial(n,j)*T[i,j])) ));U}
a(n)=G(n)[n,n]; \\ Andrew Howroyd, May 22 2017

Formula

a(n) = Sum_{i=1..n} Sum_{j=1..n} binomial(n,i)*binomial(n,j)*A262307(i,j). - Andrew Howroyd, May 22 2017
a(n) ~ 2^(n^2). - Vaclav Kotesovec, Oct 12 2017

Extensions

Terms a(7) and beyond from Andrew Howroyd, May 22 2017

A059020 Number of 2 X n checkerboards (with at least one red square) in which the set of red squares is edge connected.

Original entry on oeis.org

0, 3, 13, 40, 108, 275, 681, 1664, 4040, 9779, 23637, 57096, 137876, 332899, 803729, 1940416, 4684624, 11309731, 27304157, 65918120, 159140476, 384199155, 927538873, 2239276992, 5406092952, 13051462995, 31509019045, 76069501192, 183648021540, 443365544387
Offset: 0

Views

Author

John W. Layman, Dec 14 2000

Keywords

Comments

In other words, the number of connected (non-null) induced subgraphs in the n-ladder graph P_2 X P_n. - Eric W. Weisstein, May 02 2017
Also, the number of cycles in the grid graph P_3 X P_{n+1}. - Andrew Howroyd, Jun 12 2017

Links

Crossrefs

Row 2 of A287151 and row 2 of A231829.
See also A059021, A059524.
Cf. A000129. - Jaume Oliver Lafont, Sep 28 2009
Other sequences counting connected induced subgraphs: A020873, A059525, A286139, A286182, A286183, A286184, A286185, A286186, A286187, A286188, A286189, A286191, A285765, A285934, A286304.

Programs

  • Magma
    I:=[0, 3, 13, 40];[n le 4 select I[n] else 4*Self(n-1) - 4*Self(n-2) + Self(n-4):n in [1..30]]; // Marius A. Burtea, Aug 25 2019
  • Mathematica
    Join[{0},LinearRecurrence[{4, -4, 0, 1}, {3, 13, 40, 108}, 20]] (* Eric W. Weisstein, May 02 2017 *) (* adapted by Vincenzo Librandi, May 09 2017 *)
Table[(LucasL[n + 3, 2] - 8 n - 14)/4, {n, 0, 20}] (* Eric W. Weisstein, May 02 2017 *)

Formula

a(n) = 2*a(n-1) + a(n-2) + 4*n - 1.
From Jaume Oliver Lafont, Nov 23 2008: (Start)
a(n) = 3*a(n-1) - a(n-2) - a(n-3) + 4;
a(n) = 4*a(n-1) - 4*a(n-2) + a(n-4). (End)
G.f.: x*(3+x)/((1-2*x-x^2)*(1-x)^2). - Jaume Oliver Lafont, Sep 28 2009
Empirical observations (from Superseeker):
(1) if b(n) = a(n) + n then {b(n)} is A048777;
(2) if b(n) = a(n+3) - 3*a(n+2) - 3*a(n+1) + a(n) then {b(n)} is A052542;
(3) if b(n) = a(n+2) - 2*a(n+1) + a(n) then {b(n)} is A001333.
4*a(n) = A002203(n+3) - 8*n - 14. - Eric W. Weisstein, May 02 2017
a(n) = 3*A048776(n-1) + A048776(n-2). - R. J. Mathar, May 12 2019
E.g.f.: (1/2)*exp(x)*(-7-4*x+7*cosh(sqrt(2)*x)+5*sqrt(2)*sinh(sqrt(2)*x)). - Stefano Spezia, Aug 25 2019

A286139 Number of connected induced (non-null) subgraphs of the n X n king graph.

Original entry on oeis.org

1, 15, 388, 37196, 14765089, 24076152503, 159850328891568, 4290837646252661680, 463376724731585422732393, 200665409586497566263900755703, 347694350828123116321061347501951972, 2406781070555276417850396576804205226358828, 66481859567653621586313146932097075651519991887257
Offset: 1

Views

Author

Giovanni Resta, May 03 2017

Keywords

Links

Crossrefs

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

Programs

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

Extensions

a(10)-a(13) from Andrew Howroyd, May 20 2017

A286183 Number of connected induced (non-null) subgraphs of the antiprism graph with 2n nodes.

Original entry on oeis.org

3, 15, 60, 207, 663, 2038, 6107, 17983, 52272, 150407, 429223, 1216490, 3427635, 9609327, 26821668, 74576703, 206650167, 570877918, 1572754187, 4322192287, 11851474968, 32430381815, 88576465735, 241511251922, 657457204323, 1787147867343, 4851349002252
Offset: 1

Views

Author

Giovanni Resta, May 04 2017

Keywords

Links

Crossrefs

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

Programs

  • Mathematica
    a[n_] := Block[{g = Graph@ Flatten@ Table[{i <-> Mod[i,n]+1, n+i <-> Mod[i,n] + n+1, i <-> n + Mod[i, n] + 1, i <-> n + Mod[i-1, n] + 1}, {i, n}]}, -1 + ParallelSum[ Boole@ ConnectedGraphQ@ Subgraph[g, s], {s, Subsets@ Range[2 n]}]]; Array[a, 8]

Formula

a(n) = 8*a(n-1) - 24*a(n-2) + 34*a(n-3) - 24*a(n-4) + 8*a(n-5) - a(n-6), for n > 6 (conjectured).
a(n) = A005248(n) - 2*n + 2*n*A001906(n) (conjectured). - Eric W. Weisstein, May 08 2017
G.f.: x*(3 - 9*x + 12*x^2 - 15*x^3 + 9*x^4 - 2*x^5) / ((1 - x)^2*(1 - 3*x + x^2)^2) (conjectured). - Colin Barker, May 30 2017

Extensions

a(17)-a(27) from Andrew Howroyd, May 20 2017

A286184 Number of connected induced (non-null) subgraphs of the helm graph with 2n+1 nodes.

Original entry on oeis.org

6, 19, 56, 157, 430, 1171, 3204, 8857, 24794, 70303, 201712, 584677, 1708998, 5028715, 14873180, 44160817, 131499442, 392401207, 1172747208, 3508804477, 10506490526, 31477528579, 94344505396, 282848966857, 848161024650, 2543677767631, 7629355581344
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), A286185 (Möbius ladder), A286186 (friendship), A286187 (web), A286188 (gear), A286189 (rook), A285765 (queen).

Programs

  • Magma
    [3^n + (1+n)*2^n - n: n in [1..30]]; // Vincenzo Librandi, May 21 2017
  • Mathematica
    a[n_] := Block[{g = Graph@ Flatten@ Table[{i <-> Mod[i, n] + 1, i <-> n + Mod[i, n] + 1, i <-> 2 n + 1}, {i, n}]}, -1 + ParallelSum[ Boole@ ConnectedGraphQ@ Subgraph[g, s], {s, Subsets@ Range[2 n + 1]}]]; Array[a, 8]
Table[3^n + (1 + n) 2^n - n, {n, 30}] (* Vincenzo Librandi, May 21 2017 *)
CoefficientList[Series[(6 - 35 x + 71 x^2 - 64 x^3 + 24 x^4) / ((1-3x)(1-2x)^2(1-x)^2), {x, 0, 30}], x] (* Vincenzo Librandi, May 21 2017 *)
LinearRecurrence[{9, -31, 51, -40, 12}, {6, 19, 56, 157, 430}, 20] (* Eric W. Weisstein, May 28 2017 *)

Formula

a(n) = 3^n + (1 + n)*2^n - n.
a(n) = 9*a(n-1) - 31*a(n-2) + 51*a(n-3) - 40*a(n-4) + 12*a(n-5). - Eric W. Weisstein, May 28 2017
G.f.: x*(6 - 35*x + 71*x^2 - 64*x^3 + 24*x^4)/((1 - 3*x)*(1 - 2*x)^2*(1 - x)^2). - Vincenzo Librandi, May 21 2017
E.g.f.: exp(3*x) - x*exp(x) + exp(2*x)*(1 + 2*x) - 2. - Stefano Spezia, Aug 25 2022

Extensions

a(17)-a(27) from Andrew Howroyd, May 21 2017

A286185 Number of connected induced (non-null) subgraphs of the Möbius ladder graph with 2n nodes.

Original entry on oeis.org

3, 15, 55, 173, 511, 1451, 4019, 10937, 29371, 78055, 205679, 538149, 1399607, 3621315, 9327723, 23931633, 61186163, 155949119, 396369831, 1004904733, 2541896559, 6416348251, 16165611043, 40657256617, 102090514731, 255968753175, 640899345631, 1602640560533
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), A286186 (friendship), A286187 (web), A286188 (gear), A286189 (rook), A285765 (queen).

Programs

  • Mathematica
    a[n_] := Block[{g = CirculantGraph[2 n, {1, n}]}, -1 + ParallelSum[ Boole@ ConnectedGraphQ@ Subgraph[g, s], {s, Subsets@ Range[2 n]}]]; Array[a, 8]

Formula

a(n) = 6*a(n-1) - 11*a(n-2) + 4*a(n-3) + 5*a(n-4) - 2*a(n-5) - a(n-6), for n>6 (conjectured).
a(n) = 1/4*((1-sqrt(2))^n*(4-3*sqrt(2)*n) + (1+sqrt(2))^n*(4+3*sqrt(2)*n)) - 1 - n (conjectured). - Eric W. Weisstein, May 08 2017
a(n) = Lucas(n, 2) + 3*n*Fibonacci(n, 2) - n - 1, where Lucas(n, 2) = A002203(n) and Fibonacci(n, 2) = A000129(n) (conjectured). - Eric W. Weisstein, May 08 2017
G.f. (subject to the above conjectures. In fact all three conjectures are equivalent): (3*x-3*x^2-2*x^3-4*x^4+3*x^5-x^6)/(1-3*x+x^2+x^3)^2. - Robert Israel, May 08 2017

Extensions

a(17)-a(28) from Andrew Howroyd, May 20 2017

A286187 Number of connected induced (non-null) subgraphs of the web graph with 3n nodes.

Original entry on oeis.org

6, 33, 188, 985, 4990, 24645, 119712, 574225, 2727218, 12847821, 60115060, 279652793, 1294441894, 5965567125, 27387631368, 125308264225, 571591760602, 2600204421405, 11799376912220, 53424388364873, 241398575303374, 1088727972172389, 4901842528232304, 22034981672761649
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), A286188 (gear), A286189 (rook), A285765 (queen).

Programs

  • Mathematica
    {6, 33} ~Join~ Table[g = GraphData[{"Web", n}]; -1 + ParallelSum[Boole@ ConnectedGraphQ@ Subgraph[g, s], {s, Subsets@ Range[3 n]}], {n, 3, 6}]

Formula

Empirical g.f.: x*(6 - 39*x + 92*x^2 - 101*x^3 + 32*x^4 - 8*x^5 + 64*x^6 - 48*x^7) / ((1 - x)^2*(1 - 5*x + 2*x^2 + 4*x^3)^2). - Colin Barker, May 21 2017

Extensions

a(12)-a(24) from Andrew Howroyd, May 20 2017

A286188 Number of connected induced (non-null) subgraphs of the gear graph with 2n+1 nodes.

Original entry on oeis.org

6, 26, 76, 218, 664, 2174, 7452, 26130, 92512, 328774, 1170052, 4166106, 14836488, 52839374, 188188396, 670240802, 2387095600, 8501764310, 30279479508, 107841961962, 384084839128, 1367938434910, 4871984975932, 17351831789874, 61799465313024
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), A285765 (queen).

Programs

  • Mathematica
    a[n_] := Block[{g = Graph@ Flatten[{Table[i <-> 2 n + 1, {i, 2, 2 n, 2}], Table[i <-> Mod[i, 2 n] + 1, {i, 2 n}]}]}, -1 + ParallelSum[ Boole@ ConnectedGraphQ@ Subgraph[g, s], {s, Subsets@ Range[2 n + 1]}]]; Array[a, 8]

Formula

a(n) = 6*a(n-1) - 10*a(n-2) + 4*a(n-3) + 3*a(n-4) - 2*a(n-5), for n>5.
a(n) = A206776(n) + 4*n^2 - 2*n + 1. - Eric W. Weisstein, May 08 2017
G.f.: 2*x*(3 - 5*x - 10*x^2 - x^3 - 3*x^4) / ((1 - x)^3*(1 - 3*x - 2*x^2)). - Colin Barker, May 31 2017

Extensions

a(16)-a(25) from Andrew Howroyd, May 20 2017

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}]
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