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 1-6 of 6 results.

A323950 Number of ways to split an n-cycle into connected subgraphs, none having exactly two vertices.

Original entry on oeis.org

1, 1, 1, 2, 6, 12, 23, 44, 82, 149, 267, 475, 841, 1484, 2613, 4595, 8074, 14180, 24896, 43702, 76705, 134622, 236260, 414623, 727629, 1276917, 2240851, 3932438, 6900967, 12110373, 21252244, 37295110, 65448378, 114853920, 201554603, 353703696, 620706742
Offset: 0

Views

Author

Gus Wiseman, Feb 10 2019

Keywords

Examples

			The a(1) = 1 through a(5) = 12 partitions:
  {{1}}  {{1}{2}}  {{123}}      {{1234}}        {{12345}}
                   {{1}{2}{3}}  {{1}{234}}      {{1}{2345}}
                                {{123}{4}}      {{1234}{5}}
                                {{124}{3}}      {{1235}{4}}
                                {{134}{2}}      {{1245}{3}}
                                {{1}{2}{3}{4}}  {{1345}{2}}
                                                {{1}{2}{345}}
                                                {{1}{234}{5}}
                                                {{123}{4}{5}}
                                                {{125}{3}{4}}
                                                {{145}{2}{3}}
                                                {{1}{2}{3}{4}{5}}

Links

Crossrefs

Cf. A000325, A001610, A001680, A005251, A066982, A306351, A323950, A323951, A323952, A323954.

Programs

  • Mathematica
    cyceds[n_,k_]:=Union[Sort/@Join@@Table[1+Mod[Range[i,j]-1,n],{i,n},{j,Prepend[Range[i+k,n+i-1],i]}]];
spsu[,{}]:={{}};spsu[foo,set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@spsu[Select[foo,Complement[#,Complement[set,s]]=={}&],Complement[set,s]]]/@Cases[foo,{i,_}];
Table[Length[spsu[cyceds[n,2],Range[n]]],{n,15}]

Formula

G.f.: (x^7-3*x^6+3*x^5-2*x^4+x^3-3*x^2+3*x-1)/((x^3-x^2+2*x-1)*(x-1)^2). - Alois P. Heinz, Feb 10 2019

Extensions

More terms from Alois P. Heinz, Feb 10 2019

A323954 Regular triangle read by rows where T(n, k) is the number of ways to split an n-cycle into connected subsequences of sizes > k, n >=1, 0 <= k < n.

Original entry on oeis.org

1, 2, 1, 5, 1, 1, 12, 3, 1, 1, 27, 6, 1, 1, 1, 58, 12, 4, 1, 1, 1, 121, 22, 8, 1, 1, 1, 1, 248, 39, 13, 5, 1, 1, 1, 1, 503, 67, 22, 10, 1, 1, 1, 1, 1, 1014, 113, 36, 16, 6, 1, 1, 1, 1, 1, 2037, 188, 56, 23, 12, 1, 1, 1, 1, 1, 1, 4084, 310, 86, 35, 19, 7, 1, 1, 1, 1, 1, 1
Offset: 1

Views

Author

Gus Wiseman, Feb 10 2019

Keywords

Examples

			Triangle begins:
     1
     2    1
     5    1    1
    12    3    1    1
    27    6    1    1    1
    58   12    4    1    1    1
   121   22    8    1    1    1    1
   248   39   13    5    1    1    1    1
   503   67   22   10    1    1    1    1    1
  1014  113   36   16    6    1    1    1    1    1
  2037  188   56   23   12    1    1    1    1    1    1
  4084  310   86   35   19    7    1    1    1    1    1    1
Row 4 counts the following partitions:
  {{1234}}        {{1234}}    {{1234}}  {{1234}}
  {{1}{234}}      {{12}{34}}
  {{12}{34}}      {{14}{23}}
  {{123}{4}}
  {{124}{3}}
  {{134}{2}}
  {{14}{23}}
  {{1}{2}{34}}
  {{1}{23}{4}}
  {{12}{3}{4}}
  {{14}{2}{3}}
  {{1}{2}{3}{4}}

Links

Crossrefs

Column k = 0 is A000325. Column k = 1 is A066982. Column k = 2 is A323951. Column k = 3 is A306351.
Cf. A001610, A001680, A005251, A323950, A323951, A323952, A323953.

Programs

  • Mathematica
    cycedsprop[n_,k_]:=Union[Sort/@Join@@Table[1+Mod[Range[i,j]-1,n],{i,n},{j,i+k,n+i-1}]];
spsu[,{}]:={{}};spsu[foo,set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@spsu[Select[foo,Complement[#,Complement[set,s]]=={}&],Complement[set,s]]]/@Cases[foo,{i,_}];
Table[Length[spsu[cycedsprop[n,k],Range[n]]],{n,12},{k,0,n-1}]
  • PARI
    T(n,k) = 1 - n + sum(i=1, n\(k+1), n*binomial(n-i*k-1, i-1)/i) \\ Andrew Howroyd, Jan 19 2023

Formula

T(n,k) = 1 - n + Sum_{i=1..floor(n/(k+1))} n*binomial(n-i*k-1, i-1)/i. - Andrew Howroyd, Jan 19 2023

A323951 Number of ways to split an n-cycle into connected subgraphs, all having at least three vertices.

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 4, 8, 13, 22, 36, 56, 86, 131, 197, 294, 437, 647, 955, 1407, 2070, 3042, 4467, 6556, 9618, 14106, 20684, 30325, 44455, 65164, 95515, 139997, 205189, 300733, 440760, 645980, 946745, 1387538, 2033552, 2980332, 4367906, 6401495, 9381865, 13749810
Offset: 0

Views

Author

Gus Wiseman, Feb 10 2019

Keywords

Examples

			The a(3) = 1 through a(7) = 8 partitions:
  {{123}}  {{1234}}  {{12345}}  {{123456}}    {{1234567}}
                                {{123}{456}}  {{123}{4567}}
                                {{126}{345}}  {{1234}{567}}
                                {{156}{234}}  {{1237}{456}}
                                              {{1267}{345}}
                                              {{127}{3456}}
                                              {{1567}{234}}
                                              {{167}{2345}}

Links

Crossrefs

Cf. A000325, A001610, A001680, A005251, A066982, A306351, A323950, A323952, A323954.

Programs

  • Mathematica
    cycedsprop[n_,k_]:=Union[Sort/@Join@@Table[1+Mod[Range[i,j]-1,n],{i,n},{j,i+k,n+i-1}]];
spsu[,{}]:={{}};spsu[foo,set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@spsu[Select[foo,Complement[#,Complement[set,s]]=={}&],Complement[set,s]]]/@Cases[foo,{i,_}];
Table[Length[spsu[cycedsprop[n,2],Range[n]]],{n,15}]

Formula

G.f.: (x^7-2*x^6+x^3-3*x^2+3*x-1)/((x^3+x-1)*(x-1)^2). - Alois P. Heinz, Feb 10 2019

Extensions

More terms from Alois P. Heinz, Feb 10 2019

A323953 Regular triangle read by rows where T(n, k) is the number of ways to split an n-cycle into singletons and connected subsequences of sizes > k.

Original entry on oeis.org

1, 2, 1, 5, 2, 1, 12, 6, 2, 1, 27, 12, 7, 2, 1, 58, 23, 14, 8, 2, 1, 121, 44, 23, 16, 9, 2, 1, 248, 82, 38, 26, 18, 10, 2, 1, 503, 149, 65, 38, 29, 20, 11, 2, 1, 1014, 267, 112, 57, 42, 32, 22, 12, 2, 1, 2037, 475, 189, 90, 57, 46, 35, 24, 13, 2, 1
Offset: 1

Views

Author

Gus Wiseman, Feb 10 2019

Keywords

Examples

			Triangle begins:
     1
     2    1
     5    2    1
    12    6    2    1
    27   12    7    2    1
    58   23   14    8    2    1
   121   44   23   16    9    2    1
   248   82   38   26   18   10    2    1
   503  149   65   38   29   20   11    2    1
  1014  267  112   57   42   32   22   12    2    1
  2037  475  189   90   57   46   35   24   13    2    1
  4084  841  312  146   80   62   50   38   26   14    2    1
Row 4 counts the following connected partitions:
  {{1234}}        {{1234}}        {{1234}}        {{1}{2}{3}{4}}
  {{1}{234}}      {{1}{234}}      {{1}{2}{3}{4}}
  {{12}{34}}      {{123}{4}}
  {{123}{4}}      {{124}{3}}
  {{124}{3}}      {{134}{2}}
  {{134}{2}}      {{1}{2}{3}{4}}
  {{14}{23}}
  {{1}{2}{34}}
  {{1}{23}{4}}
  {{12}{3}{4}}
  {{14}{2}{3}}
  {{1}{2}{3}{4}}

Links

Crossrefs

First column is A000325. Second column is A323950.
Cf. A001610, A001680, A005251, A066982, A306351, A323951, A323952, A323954.

Programs

  • Mathematica
    cyceds[n_,k_]:=Union[Sort/@Join@@Table[1+Mod[Range[i,j]-1,n],{i,n},{j,Prepend[Range[i+k,n+i-1],i]}]];
spsu[,{}]:={{}};spsu[foo,set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@spsu[Select[foo,Complement[#,Complement[set,s]]=={}&],Complement[set,s]]]/@Cases[foo,{i,_}];
Table[Length[spsu[cyceds[n,k],Range[n]]],{n,10},{k,n}]
  • PARI
    T(n,k) = {1 + if(kAndrew Howroyd, Jan 19 2023

Formula

T(n,k) = 2 - n + Sum_{i=1..floor(n/k)} n*binomial(n-i*k+i-1, 2*i-1)/i for 1 <= k < n. - Andrew Howroyd, Jan 19 2023

A306357 Number of nonempty subsets of {1, ..., n} containing no three cyclically successive elements.

Original entry on oeis.org

0, 1, 3, 6, 10, 20, 38, 70, 130, 240, 442, 814, 1498, 2756, 5070, 9326, 17154, 31552, 58034, 106742, 196330, 361108, 664182, 1221622, 2246914, 4132720, 7601258, 13980894, 25714874, 47297028, 86992798, 160004702, 294294530, 541292032, 995591266, 1831177830
Offset: 0

Views

Author

Gus Wiseman, Feb 10 2019

Keywords

Comments

Cyclically successive means 1 is a successor of n.
Set partitions using these subsets are counted by A323949.

Examples

			The a(1) = 1 through a(5) = 20 stable subsets:
  {1}  {1}    {1}    {1}    {1}
       {2}    {2}    {2}    {2}
       {1,2}  {3}    {3}    {3}
              {1,2}  {4}    {4}
              {1,3}  {1,2}  {5}
              {2,3}  {1,3}  {1,2}
                     {1,4}  {1,3}
                     {2,3}  {1,4}
                     {2,4}  {1,5}
                     {3,4}  {2,3}
                            {2,4}
                            {2,5}
                            {3,4}
                            {3,5}
                            {4,5}
                            {1,2,4}
                            {1,3,4}
                            {1,3,5}
                            {2,3,5}
                            {2,4,5}

Crossrefs

Cf. A000126, A000296, A001610, A001644, A169985, A306357, A323949, A323952, A323955.

Programs

  • Mathematica
    stabsubs[g_]:=Select[Rest[Subsets[Union@@g]],Select[g,Function[ed,UnsameQ@@ed&&Complement[ed,#]=={}]]=={}&];
Table[Length[stabsubs[Partition[Range[n],3,1,1]]],{n,15}]

Formula

For n >= 3 we have a(n) = A001644(n) - 1.
From Chai Wah Wu, Jan 06 2020: (Start)
a(n) = 2*a(n-1) - a(n-4) for n > 6.
G.f.: x*(x^5 + x^4 - 2*x^3 + x + 1)/(x^4 - 2*x + 1). (End)

A323956 Triangle read by rows: T(n, k) = 1 + n * (n - k) for 1 <= k <= n.

Original entry on oeis.org

1, 3, 1, 7, 4, 1, 13, 9, 5, 1, 21, 16, 11, 6, 1, 31, 25, 19, 13, 7, 1, 43, 36, 29, 22, 15, 8, 1, 57, 49, 41, 33, 25, 17, 9, 1, 73, 64, 55, 46, 37, 28, 19, 10, 1, 91, 81, 71, 61, 51, 41, 31, 21, 11, 1, 111, 100, 89, 78, 67, 56, 45, 34, 23, 12, 1
Offset: 1

Views

Author

Gus Wiseman, Feb 10 2019

Keywords

Examples

			Triangle begins:
  n\k:   1   2   3   4   5   6   7   8   9  10  11  12
  ====================================================
    1:   1
    2:   3   1
    3:   7   4   1
    4:  13   9   5   1
    5:  21  16  11   6   1
    6:  31  25  19  13   7   1
    7:  43  36  29  22  15   8   1
    8:  57  49  41  33  25  17   9   1
    9:  73  64  55  46  37  28  19  10   1
   10:  91  81  71  61  51  41  31  21  11   1
   11: 111 100  89  78  67  56  45  34  23  12   1
   12: 133 121 109  97  85  73  61  49  37  25  13   1
  etc.

Links

Crossrefs

First column is A002061. Second column is A000290. Third column is A028387.
Cf. A000126, A001610, A001644, A006000, A169985, A306351, A323952, A323955.

Programs

  • Magma
    [[1+n*(n-k): k in [1..n]]: n in [1..12]]; // G. C. Greubel, Apr 22 2019
  • Mathematica
    Table[1+n*(n-k),{n,12},{k,n}]//Flatten
  • PARI
    {T(n,k) = 1+n*(n-k)}; \\ G. C. Greubel, Apr 22 2019
  • Sage
    [[1+n*(n-k) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Apr 22 2019

Formula

From Werner Schulte, Feb 12 2019: (Start)
G.f.: Sum_{n>0,k=1..n} T(n,k)*x^k*t^n = x*t*((1-t+2*t^2)*(1-x*t) + (1-t)*t)/((1-t)^3*(1-x*t)^2).
Row sums: Sum_{k=1..n} T(n,k) = A006000(n-1) for n > 0.
Recurrence: T(n,k) = T(n,k-1) - n for 1 < k <= n with initial values T(n,1) = n^2-n+1 for n > 0.
Recurrence: T(n,k) = T(n-1,k) + 2*n-k-1 for 1 <= k < n with initial values T(n,n) = 1 for n > 0.
(End)
Showing 1-6 of 6 results.

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