The On-Line Encyclopedia of Integer Sequences (OEIS)

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A333298

Number of canonical sequences of moves of length n for the Rubik cube puzzle using the half-turn metric.
(history; published version)

#20 by Joerg Arndt at Sat Feb 17 10:08:06 EST 2024

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reviewed

approved

#19 by Stefano Spezia at Sat Feb 17 09:16:22 EST 2024

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proposed

reviewed

#18 by Michel Marcus at Sat Feb 17 09:10:58 EST 2024

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editing

proposed

#17 by Michel Marcus at Sat Feb 17 09:10:52 EST 2024

LINKS

Tomas Rokicki, Herbert Kociemba, Morley Davidson, and John Dethridge, <a href="http://tomas.rokicki.com/rubik20.pdf">The Diameter Of The Rubik's Cube Group Is Twenty</a>, SIAM J. of Discrete Math, Vol. 27, No. 2 (2013), pp. 1082-1105.

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proposed

editing

#16 by Herbert Kociemba at Sat Feb 17 08:13:04 EST 2024

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editing

proposed

#15 by Herbert Kociemba at Sat Feb 17 07:33:05 EST 2024

FORMULA

Conjectures from _From _Colin Barker_, Mar 23 2020: (Start)

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approved

editing

Discussion

Sat Feb 17

08:09

Herbert Kociemba: The conjectures from Colin Barker are definitely true. In the outer block turn metric (OBTM) for N x N x N Rubik's Cubes I derived years ago a astonishing simple generating function for the number of canonical sequences
3/(6 - 4 (3 x + 1)^(n - 1)) - 1/2

and Colin's  formulas result from the case n=3.
For n=2 we get for example corresponding formulas
G.f.: (1+3 x)/(1-6 x)
a(n) = 6*a(n-1) for n>1
a(n)=2^(-1+n) 3^(1+n)

and for n=4
G.f: -(1/2)+3/(6-4 (1+3 x)^3)
a(n)=18*a(n-1)+54*a(n-2)+54*a(n-3) for n>3

#14 by N. J. A. Sloane at Mon Mar 23 10:52:07 EDT 2020

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editing

approved

#13 by N. J. A. Sloane at Mon Mar 23 10:52:02 EDT 2020

REFERENCES

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.

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proposed

editing

#12 by Colin Barker at Mon Mar 23 10:15:13 EDT 2020

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editing

proposed

#11 by Colin Barker at Mon Mar 23 10:14:54 EDT 2020

FORMULA

a(n) = (-(6-3*sqrt(6))^n*(-3+sqrt(6)) + (3*(2+sqrt(6)))^n*(3+sqrt(6))) / 4 for n>0.

STATUS

proposed

editing