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: Matías Benzo

Matías Benzo's wiki page.

Matías Benzo has authored 4 sequences.

A193168 Number of simple paths from (1, 1) to (n, 3) on an n X 3 grid with king moves allowed.

Original entry on oeis.org

1, 24, 235, 2922, 38169, 494596, 6375379, 82191766, 1059980385, 13670322763, 176299392634, 2273637717194, 29321902354841, 378149186319554, 4876791443282017, 62893416210079645, 811103334429880838, 10460373436120693109, 134901938876902858230, 1739759409373842739031
Offset: 1

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Author

Matías Benzo, Jul 17 2011

Keywords

Comments

Simple paths are self-avoiding walks.
Note: Number of paths from (1, 1) to (n, 2) on a n X 2 grid matches the sequence A086347.
Example: a(2) = 24 paths from (1, 1) to (2, 3)
1 [(1, 1), (2, 1), (1, 2), (1, 3), (2, 3)]
2 [(1, 1), (2, 2), (1, 3), (2, 3)]
3 [(1, 1), (2, 2), (2, 1), (1, 2), (2, 3)]
4 [(1, 1), (1, 2), (2, 1), (2, 2), (1, 3), (2, 3)]
5 [(1, 1), (2, 2), (2, 1), (1, 2), (1, 3), (2, 3)]
6 [(1, 1), (1, 2), (2, 2), (1, 3), (2, 3)]
7 [(1, 1), (1, 2), (1, 3), (2, 3)]
8 [(1, 1), (2, 2), (1, 3), (1, 2), (2, 3)]
9 [(1, 1), (2, 2), (2, 3)]
10 [(1, 1), (1, 2), (2, 3)]
11 [(1, 1), (2, 2), (1, 2), (1, 3), (2, 3)]
12 [(1, 1), (2, 1), (2, 2), (1, 3), (2, 3)]
13 [(1, 1), (2, 1), (2, 2), (1, 2), (2, 3)]
14 [(1, 1), (2, 1), (1, 2), (1, 3), (2, 2), (2, 3)]
15 [(1, 1), (1, 2), (2, 1), (2, 2), (2, 3)]
16 [(1, 1), (2, 1), (2, 2), (1, 3), (1, 2), (2, 3)]
17 [(1, 1), (2, 1), (1, 2), (2, 3)]
18 [(1, 1), (1, 2), (2, 2), (2, 3)]
19 [(1, 1), (1, 2), (1, 3), (2, 2), (2, 3)]
20 [(1, 1), (2, 1), (2, 2), (1, 2), (1, 3), (2, 3)]
21 [(1, 1), (2, 2), (1, 2), (2, 3)]
22 [(1, 1), (2, 1), (2, 2), (2, 3)]
23 [(1, 1), (2, 1), (1, 2), (2, 2), (2, 3)]
24 [(1, 1), (2, 1), (1, 2), (2, 2), (1, 3), (2, 3)]

Links

Crossrefs

Row 3 of A329118.
Cf. A086347, A140518.

Formula

Conjectures from Andrew Howroyd, Nov 05 2019: (Start)
a(n) = 12*a(n-1) + 6*a(n-2) + 50*a(n-3) + 300*a(n-4) - 194*a(n-5) - 833*a(n-6) + 352*a(n-7) + 661*a(n-8) - 84*a(n-9) - 219*a(n-10) + 72*a(n-11) + 54*a(n-12) for n > 12.
G.f.: x*(1 + 12*x - 59*x^2 - 92*x^3 + 195*x^4 + 280*x^5 + 102*x^6 - 178*x^7 - 67*x^8 + 109*x^9 - 75*x^10 - 54*x^11)/((1 + x + 9*x^2 + 24*x^3 + 9*x^4)*(1 - 13*x - 2*x^2 + 45*x^3 - 24*x^4 - 22*x^5 + 9*x^6 + 8*x^7 - 6*x^8)).
(End)

Extensions

Offset changed and more terms from Andrew Howroyd, Nov 05 2019

A193170 Number of simple paths from (1, 1) to (n, 4) on an n X 4 grid with king moves allowed.

Original entry on oeis.org

1, 116, 2922, 96371, 3764367, 150610151, 5898799685, 229260056150, 8913331166213, 346912968530666, 13505399293115778, 525732956071093698, 20464530153147155981, 796592566402543482684, 31007930721872464811580, 1207007982256603680132971, 46983727059243543906960086
Offset: 1

Views

Author

Matías Benzo, Jul 17 2011

Keywords

Links

Crossrefs

Row 4 of A329118.

Extensions

Offset changed and terms a(7) and beyond from Andrew Howroyd, Nov 05 2019

A193171 Number of simple paths from (1, 1) to (n, 5) on an n X 5 grid with king moves allowed.

Original entry on oeis.org

1, 560, 38169, 3764367, 447544629, 56182569218, 6972159602221, 860282926307373, 106058594866721427, 13079879279632244767, 1613359976756293117691, 199005867656003390145969, 24546867122140625530760786, 3027781041159666215998006805, 373467431655380156034555333637
Offset: 1

Views

Author

Matías Benzo, Jul 17 2011

Keywords

Links

Crossrefs

Row 5 of A329118.

Extensions

Offset changed and terms a(6) and beyond from Andrew Howroyd, Nov 05 2019

A193172 Number of simple paths from (1, 1) to (n, 6) on an n X 6 grid with king moves allowed.

Original entry on oeis.org

1, 2704, 494596, 150610151, 56182569218, 22132498074021, 8656506756327178, 3389938886108156202, 1331019952223152763862, 523382675043001210360214, 205904433904968682871489186, 81019053039706376245418011749, 31881826560222260172503676030065
Offset: 1

Views

Author

Matías Benzo, Jul 17 2011

Keywords

Links

Crossrefs

Row 6 of A329118.

Extensions

Offset changed and terms a(5) and beyond from Andrew Howroyd, Nov 05 2019

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