A376609 -id:A376609 - cp's OEIS frontend

A376610 a(n) is the denominator corresponding to A376609(n).

1, 5, 35, 19, 8723, 31879, 3797647, 46627015, 46174521031, 476162538587, 15682351095751655, 33959335630630535, 9299679062615813936051, 136414995946010125, 601830836638387694170497793, 26847490207486334339335997, 1114246119072163102989483761615244013, 1430040019838636422092747945537920663

Offset: 1

Hugo Pfoertner, Oct 03 2024

			1, 8/5, 72/35, 46/19, 23747/8723, 94968/31879, 12161644/3797647, 158536576/46627015, 165181795263/46174521031, ...

Hugo Pfoertner, Table of n, a(n) for n = 1..45

A376609 are the corresponding numerators.

\\ Uses function droprob from A376606 and definition of kingmoves from A376609
a376610(n) = denominator(droprob(n, kingmoves, 8))

A376606 a(n) is the numerator of the expected number of moves to reach a position outside an nXn chessboard, starting in one of the corners, when performing a random walk with unit steps on the square lattice.

Original entry on oeis.org

1, 2, 11, 10, 99, 122, 619, 4374, 187389, 482698, 11031203, 33386106, 32723853563, 139832066, 150236161755, 633573154269934, 5755694771977, 189378719187729770, 509943025510535499, 6031948951257694364778, 1044408374351599765540157091, 27891966006517951087819226666

Offset: 1

Ruediger Jehn and Hugo Pfoertner, Oct 03 2024

The moves are that of chess rook with moves of unit length or of a chess king restricted to the Von Neumann neighborhood (see A272763). The piece does not pay attention to its position and will fall off the board if it makes a move beyond the edge of the board.

			1, 2, 11/4, 10/3, 99/26, 122/29, 619/136, 4374/901, 187389/36562, 482698/89893, ...
Approximately 1, 2, 2.75, 3.333, 3.808, 4.207, 4.551, 4.855, 5.125, 5.370, 5.593, ...

Hugo Pfoertner, Table of n, a(n) for n = 1..45
Hugo Pfoertner, Plot of A376606(n)/A376607(n) vs n, using Plot 2.

A376607 are the corresponding denominators.
A376609 and A376610 are similar for a chess king visiting the Moore neighborhood.
A376736 and A376737 are similar for a chess knight.
Cf. A001263, A007764, A272763.

droprob(n,moves=[[1,0],[0,1],[0,-1],[-1,0]], nmoves=4) = {my(np=n^2+1, M=matrix(np), P=1/nmoves); for(t=1, nmoves, for( i=1, n, my(ti=i+moves[t][1]); for(j=1,n,my(tj=j+moves[t][2]); my(m=(i-1)*n+j); if(ti<1 || ti>n || tj<1 || tj>n, M[m,np]+=P, my(mt=(ti-1)*n+tj); M[m,mt]+=P)))); vecsum((1/(matid(np)-M))[,1])};
a376606(n) = numerator(droprob(n))

A376736 a(n) is the numerator of the expected number of random moves of a chess knight to reach a position outside an nXn chessboard, starting in one of the corners.

Original entry on oeis.org

1, 1, 4, 62, 269, 1766, 395497, 101338, 44125237, 227721959, 3361699348115, 483866477194862, 277887411827604127, 790848403160840410, 2785714552717079970073201, 89715505143567836216964174, 2034961072108249587083318018747, 457177774768288408431166142758841, 1085703228381446052419019696184520372520

Offset: 1

Hugo Pfoertner, Oct 03 2024

The piece does not pay attention to its position and will fall off the board if it makes a move beyond the edge of the board.

			1, 1, 4/3, 62/43, 269/167, 1766/1017, 395497/213488, 101338/51901, 44125237/21578387, 227721959/106983448, ...
Approximately 1, 1, 1.333, 1.442, 1.611, 1.736, 1.853, 1.953, 2.045, 2.129, 2.206, ...

Hugo Pfoertner, Table of n, a(n) for n = 1..45
Hugo Pfoertner, Plot of A376736(n)/A376737(n) vs n, using Plot 2.
Hugo Pfoertner, Results of a simulation of 10^9 walks on the 8X8 board.

A376737 are the corresponding denominators.
A376606 and A376607 are similar for a rook walk with unit steps.
A376609 and A376610 are similar for a chess king.
Cf. A003192, A212715, A323131, A323132, A323559, A323560.

\\ Uses function droprob from A376606
knightmoves = [[2, 1], [1, 2], [-1, 2], [-2, 1], [-2, -1], [-1, -2], [1, -2], [2, -1]];
a376736(n) = numerator(droprob(n, knightmoves, 8))

A378902 a(n) is the number of paths of a chess king on square a1 to reach a position outside an 8 X 8 chessboard after n steps.

Original entry on oeis.org

5, 6, 39, 156, 922, 5060, 31165, 196605, 1301490, 8844147, 61504902, 434181564, 3098427480, 22270496859, 160854381441, 1165549608378, 8463549600999, 61543303627788, 447926999731974, 3262077526200660, 23765765966223849, 173189189528260281, 1262299887268848702, 9201356346994752339

Offset: 1

Hugo Pfoertner, Dec 10 2024

The king visits the Moore neighborhood, and the 8 possible moves relative to its current position are E, NE, N, NW, W, SW, S, and SE.

			a(1) = 5: only the 3 moves E, NE, and N end on target squares on the chessboard, the other 5 leave the board.
a(2) = 6: the 6 combinations of step directions leaving the board in exactly 2 moves are [E,SW], [E,S], [E,SE], [N,NE], [N,E], and [N,SE].

Ruediger Jehn, Table of n, a(n) for n = 1..100

Cf. A140518, A376609, A376610, A377018.

LinearRecurrence[{9, 9, -159, -108, 810, 900, -513, -729, -27, 81}, {5, 6, 39, 156, 922, 5060, 31165, 196605, 1301490, 8844147}, 25] (* Hugo Pfoertner, May 17 2025 *)

from numpy import ones, array
P = ones((11,11),dtype=int) # transition matrix, a1=0, b1=1, c1=2, d1=3, b2=4, c2=5, d2=6, c3=7, d3=8, d4=9, off board=10
P = [[0, 2, 0, 0, 1, 0, 0, 0, 0, 0, 5, ],
     [1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 3, ],
     [0, 1, 0, 1, 1, 1, 1, 0, 0, 0, 3, ],
     [0, 0, 1, 1, 0, 1, 2, 0, 0, 0, 3, ],
     [1, 2, 2, 0, 0, 2, 0, 1, 0, 0, 0, ],
     [0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, ],
     [0, 0, 1, 2, 0, 1, 1, 1, 2, 0, 0, ],
     [0, 0, 0, 0, 1, 2, 2, 0, 2, 1, 0, ],
     [0, 0, 0, 0, 0, 1, 2, 1, 2, 2, 0, ],
     [0, 0, 0, 0, 0, 0, 0, 1, 4, 3, 0, ],
     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ]]
pop = array([1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], dtype=object) # king starts in a1
cycle = 100  # simulation period
for i in range(cycle):
    pop = pop @ P
    print(i+1, pop[10]) # Ruediger Jehn, May 17 2025

a(16) and beyond from Ruediger Jehn, May 17 2025

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A376610 a(n) is the denominator corresponding to A376609(n).

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A376606 a(n) is the numerator of the expected number of moves to reach a position outside an nXn chessboard, starting in one of the corners, when performing a random walk with unit steps on the square lattice.

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A376736 a(n) is the numerator of the expected number of random moves of a chess knight to reach a position outside an nXn chessboard, starting in one of the corners.

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A378902 a(n) is the number of paths of a chess king on square a1 to reach a position outside an 8 X 8 chessboard after n steps.

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