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.

A288518 Array read by antidiagonals: T(m,n) = number of (undirected) paths in the grid graph P_m X P_n.

Original entry on oeis.org

0, 1, 1, 3, 12, 3, 6, 49, 49, 6, 10, 146, 322, 146, 10, 15, 373, 1618, 1618, 373, 15, 21, 872, 7119, 14248, 7119, 872, 21, 28, 1929, 28917, 111030, 111030, 28917, 1929, 28, 36, 4118, 111360, 801756, 1530196, 801756, 111360, 4118, 36
Offset: 1

Views

Author

Andrew Howroyd, Jun 10 2017

Keywords

Comments

Paths of length zero are not counted here.

Examples

			Table starts:
=================================================================
m\n|  1    2      3       4         5          6            7
---|-------------------------------------------------------------
1  |  0    1      3       6        10         15           21 ...
2  |  1   12     49     146       373        872         1929 ...
3  |  3   49    322    1618      7119      28917       111360 ...
4  |  6  146   1618   14248    111030     801756      5493524 ...
5  | 10  373   7119  111030   1530196   19506257    235936139 ...
6  | 15  872  28917  801756  19506257  436619868   9260866349 ...
7  | 21 1929 111360 5493524 235936139 9260866349 343715004510 ...
...

Links

Crossrefs

Rows 2-7 are A288516, A288527, A358800, A358801, A358802, A358803.
Main diagonal is A288032.
Cf. A231829, A064298, A271465, A271592, A120443, A001184, A145157, A003763.

A288033 Number of (undirected) paths in the n X n king graph.

Original entry on oeis.org

0, 30, 5148, 6014812, 57533191444, 4956907379126694, 3954100866385811897908, 29986588563791584765930866780, 2187482261973324160097873804506155572, 1550696105068168200375810546149511240714556526
Offset: 1

Views

Author

Eric W. Weisstein, Jun 04 2017

Keywords

Comments

Paths of length zero are not counted here. - Andrew Howroyd, Jun 10 2017

Links

  • Eric Weisstein's World of Mathematics, Graph Path
  • Eric Weisstein's World of Mathematics, King Graph

Crossrefs

Cf. A236690, A288032, A288148, A234622, A158651, A140519.

Extensions

a(5)-a(10) from Andrew Howroyd, Jun 10 2017

A288516 Number of (undirected) paths in the ladder graph P_2 X P_n.

Original entry on oeis.org

1, 12, 49, 146, 373, 872, 1929, 4118, 8589, 17644, 35889, 72538, 146021, 293200, 587801, 1177278, 2356541, 4715412, 9433537, 18870210, 37744021, 75492152, 150988969, 301983206, 603972333, 1207951292, 2415909969, 4831828138, 9663665349, 19327340704
Offset: 1

Views

Author

Andrew Howroyd, Jun 10 2017

Keywords

Links

Crossrefs

Row 2 of A288518.
Cf. A288032, A137882, A287992, A020874.

Programs

  • Mathematica
    Table[18 (2^n - 1) - n (n^2 + 9 n + 41)/3, {n, 20}] (* Eric W. Weisstein, Jun 30 2017 *)
LinearRecurrence[{6, -14, 16, -9, 2}, {1, 12, 49, 146, 373}, 20] (* Eric W. Weisstein, Jun 30 2017 *)
CoefficientList[Series[(-1 - 6 x + 9 x^2 - 4 x^3)/((-1 + x)^4 (-1 + 2 x)), {x, 0, 20}], x] (* Eric W. Weisstein, Jun 30 2017 *)
  • PARI
    Vec((1+6*x-9*x^2+4*x^3)/((1-x)^4*(1-2*x))+O(x^25))
  • PARI
    a(n) = 18*(2^n - 1) - n*(n^2 + 9*n + 41)/3 \\ Charles R Greathouse IV, Jun 30 2017

Formula

a(n) = 18*(2^n - 1) - n*(n^2 + 9*n + 41)/3. - Eric W. Weisstein, Jun 30 2017
a(n) = 6*a(n-1)-14*a(n-2)+16*a(n-3)-9*a(n-4)+2*a(n-5) for n > 5.
G.f.: x*(1+6*x-9*x^2+4*x^3)/((1-x)^4*(1-2*x)).
a(n) = 18*(2^n-1) - (41*n)/3 - 3*n^2 - n^3/3. - Colin Barker, Jun 11 2017

A236753 Number of simple (non-intersecting) directed paths on the grid graph P_n X P_n.

Original entry on oeis.org

1, 28, 653, 28512, 3060417, 873239772, 687430009069, 1532025110398168, 9829526954625359697, 183563561823425961932572, 10056737067604248527218979485, 1626248896102138091401810358337184
Offset: 1

Views

Author

Jaimal Ichharam, Jan 30 2014

Keywords

Comments

This is the number of directed paths on P_n X P_n of any length and also includes one zero length path per vertex. - Andrew Howroyd, May 27 2017

Examples

			For n=2 there are 4 zero length paths (one for each vertex), 8 paths with 1 one edge, 8 paths with 2 edges and 8 paths with 3 edges, so a(2)=28. - _Andrew Howroyd_, May 27 2017

Crossrefs

Cf. A236690 (includes diagonal edges).
Cf. A288032, A007764, A121785, A120443.

Formula

a(n) = 2*A288032(n) + n^2. - Andrew Howroyd, Jun 10 2017

Extensions

a(6) corrected and a(8) added from Jaimal Ichharam, Feb 13 2014
a(6)-a(8) corrected and a(9)-a(12) from Andrew Howroyd, May 27 2017
Showing 1-4 of 4 results.

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