Ricardo Bittencourt - cp's OEIS frontend

User: Ricardo Bittencourt

Ricardo Bittencourt has authored 2 sequences.

A300665 Number of n-step paths made by a chess king, starting from the corner of an infinite chessboard, and never revisiting a cell.

1, 3, 15, 75, 391, 2065, 11091, 60215, 330003, 1820869, 10103153, 56313047, 315071801, 1768489771, 9953853677, 56158682949, 317505199769, 1798402412539

Offset: 0

Ricardo Bittencourt, Mar 10 2018

All terms are odd.

			For n=2, the a(2)=15 paths are:
.
.    0 . .     0 . .     0 . .     0 2 .     0 . .
.    |         |         |         |/         \
.    1 . .     1 . .     1-2 .     1 . .     2-1 .
.    |          \
.    2 . .     . 2 .     . . .     . . .     . . .
.
.    0 . .     0 . .     0 . .     0 . .     0 . 2
.     \         \         \         \         \ /
.    . 1 .     . 1 .     . 1 .     . 1-2     . 1 .
.     /          |          \
.    2 . .     . 2 .     . . 2     . . .     . . .
.
.    0 2 .     0-1 .     0-1 .     0-1 .     0-1-2
.     \|        /          |          \
.    . 1 .     2 . .     . 2 .     . . 2     . . .
.
.    . . .     . . .     . . .     . . .     . . .

Ricardo Bittencourt, C++ code, GitHubGist.
Ricardo Bittencourt, Go code, GitHubGist.
OEIS Wiki: Self-avoiding walks

A038373 is the same process, but using only horizontal and vertical moves.

Cf. A005773, A272469, A272763, A272773.

Go
```
(see GitHubGist link)
```

next[x_]:=Map[x + #&, Tuples[{-1, 0, 1}, 2]]
valid[s_]:=Select[next[s[[-1]]], 0<=#[[1]] && 0<=#[[2]] && FreeQ[s,#] &]
nextpath[p_]:=Outer[Append,{p},valid[p],1]
iterate[p_]:=Flatten[Map[nextpath, p], 2]
Table[Length[Nest[iterate, {{{0,0}}}, n-1]], {n,1,7}]

a(n) = A272469(n) + 2*A005773(n+1) - 1 for n > 0. - Andrey Zabolotskiy, Mar 12 2018

A226226 Number of alignments of n points with no singleton cycles.

Original entry on oeis.org

1, 0, 1, 2, 12, 64, 470, 3828, 36456, 387840, 4603392, 60061440, 855664656, 13207470912, 219609303888, 3912940891104, 74377769483520, 1502277409668096, 32130095812624512, 725400731911792896, 17240044059713320704, 430231117562438446080, 11248105572520779755520

Offset: 0

Ricardo Bittencourt, May 31 2013

nonn

a(n) is the number of labeled sequences of cycles, where no cycle has size 1.

			For n=4, the a(4)=12 alignments with no singletons are: 1234, 1243, 1324, 1342, 1423, 1432, 12|34, 13|24, 14|23, 23|14, 24|13, 34|12.

P. Flajolet and R. Segdewick, Analytic Combinatorics, Cambridge University Press, 2009, page 119

Vincenzo Librandi, Table of n, a(n) for n = 0..200

The alignments with singletons included are given by A007840.

Range[0, 50]! CoefficientList[ Series[(1 + z - Log[1/(1 - z)])^(-1), {z, 0, 50}], z]

x='x+O('x^66); Vec(serlaplace(1/(1+x-log(1/(1-x))))) \\ Joerg Arndt, Jun 01 2013

E.g.f.: 1/(1+x-log(1/(1-x)))
a(n) ~ n!*c/(1-c)^(n+2), where c = -LambertW(-exp(-2)) = 0.158594339563... - Vaclav Kotesovec, Jun 02 2013
a(0) = 1; a(n) = Sum_{k=0..n-2} binomial(n,k) * (n-k-1)! * a(k). - Ilya Gutkovskiy, Apr 26 2021

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User: Ricardo Bittencourt

A300665 Number of n-step paths made by a chess king, starting from the corner of an infinite chessboard, and never revisiting a cell.

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