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 1-5 of 5 results.

A181394 Summed lengths of all nonintersecting rook paths on a 3 x n board.

Original entry on oeis.org

2, 14, 64, 284, 1206, 4882, 19060, 72588, 271548, 1001964, 3656480, 13223348, 47461350, 169263658, 600355808, 2119297852, 7450253362, 26095036854, 91102304600, 317127751352, 1101029901244, 3813576283628, 13180379580636, 45463936339816
Offset: 1

Views

Author

David Scambler, Oct 17 2010

Keywords

Comments

Paths are self-avoiding from one corner to the diagonally opposite corner.

Examples

			E.g. s(2) {RRD, DRR, RDR, DRURD} 3+3+3+5 = 14.

Links

Crossrefs

Enumeration of these paths is A006192, related sequences A181395, A181396, A181397, A181398, A181399.

Programs

  • Maple
    a:= n-> (Matrix(8, (i,j)->if i+1=j then 1 elif i=8 then [-1, -4, 2, 4, -23, 28, -22, 8][j] else 0 fi)^n. <<0, 2, 14, 64, 284, 1206, 4882, 19060>>)[1, 1]: seq (a(n), n=1..24);  # Alois P. Heinz, Nov 26 2010
  • Mathematica
    LinearRecurrence[{8, -22, 28, -23, 4, 2, -4, -1}, {2, 14, 64, 284, 1206, 4882, 19060, 72588}, 24] (* Jean-François Alcover, Jan 03 2022 *)

Formula

G.f.: 2*x*(1 - x - 2*x^2 + 12*x^3 - 2*x^4 + 2*x^5 - 2*x^6)/((1 - x + x^2)^2*(1 - 3*x - x^2)^2). - Alois P. Heinz, Nov 26 2010, modified Andrew Howroyd, Jan 06 2020
Asymptotics: a(n) ~ ((3/4-sqrt(13)/52)*n-1/4-sqrt(13)/52)*((sqrt(13)+3)/2)^n. - Vaclav Kotesovec, Aug 31 2012

Extensions

More terms from Alois P. Heinz, Nov 26 2010

A181396 Summed lengths of nonintersecting rook paths on a 5 X n board.

Original entry on oeis.org

4, 104, 1206, 13132, 148432, 1692480, 18893254, 205592002, 2192518424, 23040437586, 239444000256, 2466260663869, 25213885065100, 256147799888040, 2588077377850470, 26026306944344453, 260646721044701192, 2600801818956016811, 25867424061321255322, 256529866716776061643
Offset: 1

Views

Author

David Scambler, Oct 17 2010

Keywords

Comments

Paths are self-avoiding from one corner to the diagonally opposite corner.

Links

Crossrefs

Row 5 of A181399.
Enumeration of these paths is A007787, related sequences A181394, A181395, A181397, A181398.

Extensions

Terms a(8) and beyond from Andrew Howroyd, Jan 06 2020

A181397 Summed lengths of nonintersecting rook paths on a 6 X n board.

Original entry on oeis.org

5, 256, 4882, 88608, 1692480, 32871240, 629907345, 11836974284, 218772951624, 3990894156672, 72039322124013, 1289057677580204, 22896767645452466, 404155803725865164, 7095412544059877973, 123985136981584276480, 2157631059961863479079, 37411816576772657794784
Offset: 1

Views

Author

David Scambler, Oct 17 2010

Keywords

Comments

Paths are self-avoiding from one corner to the diagonally opposite corner.

Links

Crossrefs

Row 6 of A181399.
Related sequences A181394, A181395, A181396, A181398.

Extensions

Terms a(7) and beyond from Andrew Howroyd, Jan 06 2020

A181399 Summed lengths of nonintersecting rook paths on an n X k board (square array by antidiagonals).

Original entry on oeis.org

0, 1, 1, 2, 4, 2, 3, 14, 14, 3, 4, 40, 64, 40, 4, 5, 104, 284, 284, 104, 5, 6, 256, 1206, 1912, 1206, 256, 6, 7, 608, 4882, 13132, 13132, 4882, 608, 7, 8, 1408, 19060, 88608, 148432, 88608, 19060, 1408, 8, 9, 3200, 72588, 577727, 1692480, 1692480, 577727, 72588, 3200, 9
Offset: 1

Views

Author

David Scambler, Oct 17 2010

Keywords

Comments

Paths are self-avoiding from one corner to the diagonally opposite corner.

Examples

			Array starts:
======================================================
n\k| 1   2     3      4        5        6        7
---|--------------------------------------------------
1  | 0   1     2      3        4        5        6 ...
2  | 1   4    14     40      104      256      608 ...
3  | 2  14    64    284     1206     4882    19060 ...
4  | 3  40   284   1912    13132    88608   577727 ...
5  | 4 104  1206  13132   148432  1692480 18893254 ...
6  | 5 256  4882  88608  1692480 32871240 ...
7  | 6 608 19060 577727 18893254 ...
...

Links

Crossrefs

Related sequences: A181394 (3 X n), A181395 (4 X n), A181396 (5 X n), A181397 (6 X n), A181398 (n X n).
The number of these paths is given in A064298.

Extensions

a(55) and a(60) in b-file corrected by Andrew Howroyd, Feb 23 2018

A181398 Summed lengths of nonintersecting rook paths on an n X n board.

Original entry on oeis.org

0, 4, 64, 1912, 148432, 32871240, 20744801272, 37559607987584, 198461813970383240, 3100218604533034920976, 144170284249598867501946128, 20038589080922552365824887715432, 8349298401437323416105571304163524760, 10453318954664242363219930783045003561923656
Offset: 1

Views

Author

David Scambler, Oct 17 2010

Keywords

Comments

Paths are self-avoiding from one corner to the diagonally opposite corner.

Examples

			For 2 x 2: right+down (length 2), down+right (length 2), s(2) = 2+2 = 4

Crossrefs

Main diagonal of A181399.
Related sequences A181394, A181395, A181396, A181397.

Extensions

a(7)-a(14) from Andrew Howroyd, Jan 07 2020
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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