A080601 -id:A080601 - cp's OEIS frontend

A333298 Number of canonical sequences of moves of length n for the Rubik cube puzzle using the half-turn metric.

1, 18, 243, 3240, 43254, 577368, 7706988, 102876480, 1373243544, 18330699168, 244686773808, 3266193870720, 43598688377184, 581975750199168, 7768485393179328, 103697388221736960, 1384201395738071424, 18476969736848122368, 246639261965462754048, 3292256598848819251200

Offset: 0

N. J. A. Sloane, Mar 23 2020

nonn

Rokicki, Tomas. Thirty years of computer cubing: The search for God's number. 2014. Reprinted in "Barrycades and Septoku: Papers in Honor of Martin Gardner and Tom Rogers", ed. Thane Plambeck and Tomas Rokicki, MAA Press, 2020, pp. 79-98.
Rokicki, T., Kociemba, H., Davidson, M., & Dethridge, J. (2014). The diameter of the rubik's cube group is twenty. SIAM REVIEW, 56(4), 645-670. Table 5.1 gives terms 0 through 20.

Tomas Rokicki, Herbert Kociemba, Morley Davidson, and John Dethridge, The Diameter Of The Rubik's Cube Group Is Twenty, SIAM J. of Discrete Math, Vol. 27, No. 2 (2013), pp. 1082-1105.

Cf. A080601, A080602, A333299.

From Colin Barker, Mar 23 2020: (Start)
G.f.: (1 + 3*x)^2 / (1 - 12*x - 18*x^2).
a(n) = 12*a(n-1) + 18*a(n-2) for n>2.
a(n) = (-(6-3*sqrt(6))^n*(-3+sqrt(6)) + (3*(2+sqrt(6)))^n*(3+sqrt(6))) / 4 for n>0.
(End)

A333299 Number of canonical sequences of moves of length n for the Rubik's cube puzzle using the quarter-turn metric.

Original entry on oeis.org

1, 12, 114, 1068, 10011, 93840, 879624, 8245296, 77288598, 724477008, 6791000856, 63656530320, 596694646092, 5593212493440, 52428869944896, 491450379709824, 4606688566257048, 43181530471120320, 404768967341615520, 3794166513675844032, 35565225338407615152

Offset: 0

N. J. A. Sloane, Mar 23 2020

Rokicki, Tomas. Thirty years of computer cubing: The search for God's number. 2014. Reprinted in "Barrycades and Septoku: Papers in Honor of Martin Gardner and Tom Rogers", ed. Thane Plambeck and Tomas Rokicki, MAA Press, 2020, pp. 79-98. Table 9.5 gives terms 0 through 18.

Cf. A080601, A080602, A333298.

CoefficientList[Series[1 + 3a/(1 - 2a) /. a -> (1 + x)^4 - 1, {x, 0, 100}], x] (* Ben Whitmore, Dec 30 2024 *)

Conjectures from Colin Barker, Mar 23 2020: (Start)
G.f.: (1 + x)^4 / (1 - 8*x - 12*x^2 - 8*x^3 - 2*x^4).
a(n) = 8*a(n-1) + 12*a(n-2) + 8*a(n-3) + 2*a(n-4) for n>4.
(End)
The above conjectured formulas are true. - Ben Whitmore, Dec 30 2024

More terms from Ben Whitmore, Dec 30 2024

A080616 Consider 3 X 3 X 3 Rubik cube, but only allow the moves T2, F2, R2; sequence gives number of positions that are exactly n moves from the start.

Original entry on oeis.org

1, 3, 6, 12, 24, 48, 93, 180, 315, 489, 604, 522, 250, 42, 3

Offset: 0

N. J. A. Sloane, Feb 26 2003

Not every position can be reached using this restricted set of moves. Total number of positions that can be reached = 2592.

Mark Longridge, God's Algorithm Calculations for Rubik's Cube...

Cf. A080601, A080614, etc.

A080617 Consider a 3 X 3 X 3 Rubik cube, but only allow the moves M_R, D; sequence gives number of positions that are exactly n moves from the start.

Original entry on oeis.org

1, 4, 10, 24, 58, 140, 338, 816, 1909, 4296, 8893, 17160, 28891, 37996, 37678, 27186, 13051, 4128, 1199, 372, 122, 36, 10, 2

Offset: 0

N. J. A. Sloane, Feb 26 2003

From Ben Whitmore, Dec 27 2024: (Start)
An alternative description is number of positions of the  subgroup of the Rubik cube at a distance of n moves from the solved state, in the slice-quarter-turn metric.
The slice-quarter-turn metric counts quarter-turns of outer faces and inner slices as 1 move, and half-turns of outer faces and inner slices as 2 moves.
The M slice is the middle layer between the left and the right faces, and an M move is a clockwise move of this slice when viewed from the left face.
The total number of positions is 6! * 2^5 * 4 * 2 = 184320.
The two positions requiring 23 moves can be reached by applying the following algorithm, or its inverse, to a solved cube: U M U M U M U M U' M U M' U M U M' U M U' M' U2 M. (End)

Mark Longridge, God's Algorithm Calculations for Rubik's Cube...

Cf. A080601, A080614, etc.

a(11)-a(23) added by Ben Whitmore, Dec 27 2024

A080618 Consider 3 X 3 X 3 Rubik cube, but only allow a certain set of 5 moves; sequence gives number of positions that are exactly n moves from the start.

Original entry on oeis.org

1, 10, 77, 584, 4434, 33664, 255320, 1933936, 14635503, 110685344

Offset: 0

N. J. A. Sloane, Feb 26 2003

Jerry Bryan, posting to Cube Lovers List, Oct 09 1995.

Alan Bawden, Cube Lovers Archive, Part 17
Mark Longridge, God's Algorithm Calculations for Rubik's Cube...

Cf. A080601, A080614, etc.

A080619 Consider 3 X 3 X 3 Rubik cube, but only allow the slice group to act; sequence gives number of positions that are exactly n moves from the start.

Original entry on oeis.org

1, 6, 27, 120, 287, 258, 69

Offset: 0

N. J. A. Sloane, Feb 26 2003

Not every position can be reached using this restricted set of moves. Total number of positions that can be reached = 768.

Mark Longridge, posting to Cube Lovers List, Apr 14 1995.

Alan Bawden, Cube Lovers Archive, Part 15
Mark Longridge, God's Algorithm Calculations for Rubik's Cube...

Cf. A080601, A080614, etc.

A080620 Consider 3 X 3 X 3 Rubik cube, but only allow the anti-slice group to act; sequence gives number of positions that are exactly n moves from the start.

Original entry on oeis.org

1, 6, 27, 120, 423, 1098, 1770, 1650, 851, 198

Offset: 0

N. J. A. Sloane, Feb 26 2003

Not every position can be reached using this restricted set of moves. Total number of positions that can be reached = 6144. This is for 2q or anti-slice moves.

Mark Longridge, posting to Cube Lovers List, Apr 14 1995 and May 21 1995.

Alan Bawden, Cube Lovers Archive, Part 15
Alan Bawden, Cube Lovers Archive, Part 16
Mark Longridge, God's Algorithm Calculations for Rubik's Cube...

Cf. A080601, A080614, etc.

A080621 Consider 3 X 3 X 3 Rubik cube, but only allow the double anti-slice group to act; sequence gives number of positions that are exactly n moves from the start.

Original entry on oeis.org

1, 9, 51, 265, 864, 1785, 2017, 1008, 144

Offset: 0

N. J. A. Sloane, Feb 26 2003

Not every position can be reached using this restricted set of moves. Total number of positions that can be reached = 6144.

Mark Longridge, posting to Cube Lovers List, Apr 14 1995 and May 21 1995.

Alan Bawden, Cube Lovers Archive, Part 15
Alan Bawden, Cube Lovers Archive, Part 16
Mark Longridge, God's Algorithm Calculations for Rubik's Cube...

Cf. A080601, A080614, etc.

A080622 Consider 3 X 3 X 3 Rubik cube, but only allow the slice group to act; sequence gives number of positions that are exactly n moves from the start, up to equivalence under the full group of order 48 of the cube.

Original entry on oeis.org

1, 1, 2, 6, 16, 15, 9

Offset: 0

N. J. A. Sloane, Feb 26 2003

Not every position can be reached using this restricted set of moves. Number of inequivalent positions that can be reached = 50.

Mark Longridge, God's Algorithm Calculations for Rubik's Cube...

Cf. A080601, A080614, etc.

A080623 Consider 3 X 3 X 3 Rubik cube, but only allow the double slice group to act; sequence gives number of positions that are exactly n moves from the start, up to equivalence under the full group of order 48 of the cube.

Original entry on oeis.org

1, 2, 4, 15, 25, 3

Offset: 0

N. J. A. Sloane, Feb 26 2003

Not every position can be reached using this restricted set of moves. Number of inequivalent positions that can be reached = 50.

Mark Longridge, God's Algorithm Calculations for Rubik's Cube...

Cf. A080601, A080614, etc.

cp's OEIS Frontend

A333298 Number of canonical sequences of moves of length n for the Rubik cube puzzle using the half-turn metric.

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A333299 Number of canonical sequences of moves of length n for the Rubik's cube puzzle using the quarter-turn metric.

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A080616 Consider 3 X 3 X 3 Rubik cube, but only allow the moves T2, F2, R2; sequence gives number of positions that are exactly n moves from the start.

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A080617 Consider a 3 X 3 X 3 Rubik cube, but only allow the moves M_R, D; sequence gives number of positions that are exactly n moves from the start.

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A080618 Consider 3 X 3 X 3 Rubik cube, but only allow a certain set of 5 moves; sequence gives number of positions that are exactly n moves from the start.

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A080619 Consider 3 X 3 X 3 Rubik cube, but only allow the slice group to act; sequence gives number of positions that are exactly n moves from the start.

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A080620 Consider 3 X 3 X 3 Rubik cube, but only allow the anti-slice group to act; sequence gives number of positions that are exactly n moves from the start.

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A080621 Consider 3 X 3 X 3 Rubik cube, but only allow the double anti-slice group to act; sequence gives number of positions that are exactly n moves from the start.

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A080622 Consider 3 X 3 X 3 Rubik cube, but only allow the slice group to act; sequence gives number of positions that are exactly n moves from the start, up to equivalence under the full group of order 48 of the cube.

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A080623 Consider 3 X 3 X 3 Rubik cube, but only allow the double slice group to act; sequence gives number of positions that are exactly n moves from the start, up to equivalence under the full group of order 48 of the cube.

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