A376737
a(n) is the denominator corresponding to A376736(n).
Original entry on oeis.org
1, 1, 3, 43, 167, 1017, 213488, 51901, 21578387, 106983448, 1524134453409, 212520825762723, 118603854051948819, 328857354494351169, 1131079058617495914656969, 35636007162246675331778279, 792054341291879335697891524219, 174615658931159537184638645409827, 407432375846003705593053861468274012573
Offset: 1
1/1, 1/1, 4/3, 62/43, 269/167, 1766/1017, 395497/213488, 101338/51901, ...
A376736 are the corresponding numerators.
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
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, ...
A376607 are the corresponding denominators.
A376609 and
A376610 are similar for a chess king visiting the Moore neighborhood.
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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))
A376609
a(n) is the numerator of the expected number of random moves of a chess king to reach a position outside an nXn chessboard, starting in one of the corners.
Original entry on oeis.org
1, 8, 72, 46, 23747, 94968, 12161644, 158536576, 165181795263, 1779861954248, 60921563004721184, 136512657826472304, 38548316743830620183051, 581371653539561314, 2630585854108441990301102856, 120104329127347395409698056, 5092493809189909792181005355935991197, 6666722670813237580783418910187983288
Offset: 1
1, 8/5, 72/35, 46/19, 23747/8723, 94968/31879, 12161644/3797647, 158536576/46627015, 165181795263/46174521031, ...
Approximately 1, 1.6, 2.057, 2.421, 2.722, 2.979, 3.202, 3.400, 3.577, 3.738, ...
A376610 are the corresponding denominators.
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\\ Uses function droprob from A376606
kingmoves = [[1, 0], [0, 1], [0, -1], [-1, 0], [-1, -1], [-1, 1], [1, -1], [1, 1]];
a376609(n) = numerator(droprob(n,kingmoves,8))
A377018
a(n) is the number of paths of a knight on square a1 to reach a position outside an 8 X 8 chessboard after n steps.
Original entry on oeis.org
6, 4, 32, 108, 880, 4420, 29560, 167068, 1041440, 6128772, 37298680, 222571260, 1343492128, 8055277508, 48487848472, 291196932444, 1751154444320, 10522542700868, 63258161555448, 380185630909692, 2285299322957920, 13735692739737604, 82562224208247576, 496247752160871132
Offset: 1
a(2) = 4. Starting on square a1 there are 4 paths for a knight to leave the chess board in two steps: b3-z2, b3-z4, c2-b0 and c2-d0 (z being the file left of the a-file).
- Ruediger Jehn, Table of n, a(n) for n = 1..30
- Index entries for linear recurrences with constant coefficients, signature (3,27,-29,-162,42,276,16,-96).
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LinearRecurrence[{3, 27, -29, -162, 42, 276, 16, -96}, {6, 4, 32, 108, 880, 4420, 29560, 167068}, 24] (* Hugo Pfoertner, Oct 16 2024 *)
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Vec(2*(3 - 7*x - 71*x^2 + 39*x^3 + 390*x^4 + 94*x^5 - 484*x^6 - 240*x^7)/(1 - 3*x - 27*x^2 + 29*x^3 + 162*x^4 - 42*x^5 - 276*x^6 - 16*x^7 + 96*x^8) + O(x^30)) \\ Andrew Howroyd, Oct 16 2024
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