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A060572 Tower of Hanoi: the optimal way to move an even number of disks from peg 0 to peg 2 or an odd number from peg 0 to peg 1 is on move n to move disk A001511 from peg A060571 to peg A060572 (here). 7

%I #64 Dec 16 2021 18:40:46

%S 1,2,2,1,0,1,1,2,2,0,0,2,1,2,2,1,0,1,1,0,2,0,0,1,1,2,2,1,0,1,1,2,2,0,

%T 0,2,1,2,2,0,0,1,1,0,2,0,0,2,1,2,2,1,0,1,1,2,2,0,0,2,1,2,2,1,0,1,1,0,

%U 2,0,0,1,1,2,2,1,0,1,1,0,2,0,0,2,1,2,2,0,0,1,1,0,2,0,0,1,1,2,2,1,0,1,1,2,2

%N Tower of Hanoi: the optimal way to move an even number of disks from peg 0 to peg 2 or an odd number from peg 0 to peg 1 is on move n to move disk A001511 from peg A060571 to peg A060572 (here).

%C If written in a fractal pattern of 4 X 4 squares, skipping the first square, going right then down then right then down, etc.:

%C X122 1011 ...

%C 1011 0200

%C 2200 1122

%C 2122 1011

%C a number of patterns become apparent. Most notably the central diagonal going from the X down and to the right, when the 1's and 2's are reversed, gives the sequence A060571. When the same process is applied to A060571, this sequence emerges. - Donald Sampson (marsquo(AT)hotmail.com), Dec 01 2003

%H Gary W. Adamson, <a href="/A060572/a060572_1.txt">Comments on A060572</a>

%H J.-P. Allouche, D. Astoorian, J. Randall, and J. Shallit, <a href="http://www.jstor.org/stable/2974693">Morphisms, squarefree strings, and the Tower of Hanoi puzzle</a>, Amer. Math. Monthly 101 (1994), 651-658.

%H <a href="/index/To#Hanoi">Index entries for sequences related to Towers of Hanoi</a>

%F a(n) = A060571(n) - (-1)^A001511(n) mod 3.

%F If n > 2^A001511(n) then a(n) = a(n-2^A001511(n)) - (-1)^A001511(n) mod 3, otherwise a(k) = -(-1)^A001511(n) mod 3.

%F a(n) = A001511(n)-th digit from right of A055662(n).

%F If a(n)=0 then a(2n)=0, If a(n)=1 then a(2n)=2, If a(n)=2 then a(2n)=1, Thus a(n)=a(4n). - Donald Sampson (marsquo(AT)hotmail.com), Dec 01 2003

%F a(5n) = A060571(n) with the 1's and 2s reversed. - Donald Sampson (marsquo(AT)hotmail.com), Dec 08 2003

%e Start by moving first disk (from peg 0) to peg 1, second disk (from peg 0) to peg 2, first disk (from peg 1) to peg 2, etc., so sequence starts 1,2,2,...

%o (PARI) a(n) = (- (-1)^valuation(n,2) - n)%3; \\ _Kevin Ryde_, Aug 07 2021

%Y Cf. A001511, A055662, A060571, A060573, A060574, A060575.

%K easy,nonn

%O 1,2

%A _Henry Bottomley_, Apr 03 2001

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Last modified March 29 06:57 EDT 2024. Contains 371265 sequences. (Running on oeis4.)