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.

User: Jaimal Ichharam

Jaimal Ichharam's wiki page.

Jaimal Ichharam has authored 3 sequences.

A245251 Triangle T(a,b) read by rows: sum_{k=1..a-1} floor(k*b/a).

Original entry on oeis.org

0, 0, 1, 0, 1, 3, 0, 2, 3, 6, 0, 2, 4, 6, 10, 0, 3, 6, 8, 10, 15, 0, 3, 6, 9, 12, 15, 21, 0, 4, 7, 12, 14, 18, 21, 28, 0, 4, 9, 12, 16, 21, 24, 28, 36, 0, 5, 9, 14, 20, 23, 27, 32, 36, 45, 0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 55, 0, 6, 12, 18, 22, 30, 33, 40, 45, 50, 55, 66, 0, 6, 12, 18, 24, 30, 36, 45
Offset: 1

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Author

Jaimal Ichharam, Jul 15 2014

Keywords

Examples

			  0;
  0   1;
  0   1   3;
  0   2   3   6;
  0   2   4   6  10;
  0   3   6   8  10  15;
  0   3   6   9  12  15  21;
  0   4   7  12  14  18  21  28;
  0   4   9  12  16  21  24  28  36;
  0   5   9  14  20  23  27  32  36  45;

Crossrefs

Cf. A000217 (right diagonal).

Programs

  • Mathematica
    Table[Sum[Floor[(k b)/a], {k, a-1}], {a, 20}, {b, a}] // Flatten

Formula

T(a, b) = (a*b-a-b+gcd(a,b))/2.
T(n, n) = n*(n-1)/2.

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

A236690 Number of simple directed paths on an n X n king graph.

Original entry on oeis.org

1, 64, 10305, 12029640, 115066382913, 9913814758253424, 7908201732771623795865, 59973177127583169531861733624, 4374964523946648320195747609012311225, 3101392210136336400751621092299022481429113152
Offset: 1

Views

Author

Jaimal Ichharam, Jan 30 2014

Keywords

Comments

This is the number (assuming each character is unique) of strings that are possible in an n X n Boggle grid.
This sequence gives the number of directed paths. Paths may start and end on any vertex and may be of any length. Paths of length zero are counted here. - Andrew Howroyd, May 12 2017

Links

  • Eric Weisstein's World of Mathematics, King Graph

Crossrefs

Cf. A236753 for a simpler version of the problem without diagonal edges.
Cf. A288033, A140518, A158651.

Formula

a(n) = 2*A288033(n) + n^2. - Andrew Howroyd, Jun 10 2017

Extensions

a(5) from Jaimal Ichharam, Feb 12 2014
Named edited and a(6)-a(10) from Andrew Howroyd, May 12 2017

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