A060586 -id:A060586 - cp's OEIS frontend

A060583 A ternary code related to the Tower of Hanoi.

0, 2, 1, 7, 6, 8, 5, 4, 3, 23, 22, 21, 18, 20, 19, 25, 24, 26, 16, 15, 17, 14, 13, 12, 9, 11, 10, 70, 69, 71, 68, 67, 66, 63, 65, 64, 54, 56, 55, 61, 60, 62, 59, 58, 57, 77, 76, 75, 72, 74, 73, 79, 78, 80, 50, 49, 48, 45, 47, 46, 52, 51, 53, 43, 42, 44, 41, 40, 39, 36, 38, 37

Offset: 0

Henry Bottomley, Apr 04 2001

Write n in base 3, then (working from left to right) if the k-th digit of n is equal to the corresponding digit to the left of the k-th digit of a(n) then this is the k-th digit of a(n), otherwise the k-th digit of a(n) is the element of {0,1,2} which has not just been compared, then read result as a base 3 number.

			a(46) = 76 since 43 = 1201_3; this gives a first digit of 2(=3-1-0), a second digit of 2(=2=2), a third digit of 1(=3-2-0) and a fourth digit of 1(=1=1); 2211_3 = 76.

Cf. A060586, A060587 (inverse).

a(n) = 3*a(floor(n/3)) + ((-a(floor(n/3))-n) mod 3) = 3*a(floor(n/3)) + A060582(n) with a(0)=0.

A060590 Numerator of the expected time to finish a random Tower of Hanoi problem with n disks using optimal moves.

Original entry on oeis.org

0, 2, 2, 14, 10, 62, 42, 254, 170, 1022, 682, 4094, 2730, 16382, 10922, 65534, 43690, 262142, 174762, 1048574, 699050, 4194302, 2796202, 16777214, 11184810, 67108862, 44739242, 268435454, 178956970, 1073741822, 715827882, 4294967294

Offset: 0

Henry Bottomley, Apr 05 2001

			a(2)=2 since there are nine equally likely possibilities, with times required of 0,1,1,2,2,3,3,3,3 giving an average of 18/9 = 2/1.

Harry J. Smith, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (0,5,0,-4).
Index entries for sequences related to Towers of Hanoi

Denominator is A010684(n). Cf. A007798, A060586, A060589, A020988 (even bisection).

a(n)={2*(2^n - 1)*(2 - (-1)^n)/3} \\ Harry J. Smith, Jul 07 2009

a(n) = 2*(2^n - 1)*(2 - (-1)^n)/3.
a(2n) = A020988(n-1).
From Ralf Stephan, Mar 07 2003: (Start)
G.f.: (4*x^3+2*x^2+2*x)/(4*x^4-5*x^2+1).
a(n+4) = 5*a(n+2) - 4*a(n). (End)

A060589 a(n) = 2(2^n-1)3^(n-1).

Original entry on oeis.org

0, 2, 18, 126, 810, 5022, 30618, 185166, 1115370, 6705342, 40271418, 241746606, 1450833930, 8706066462, 52239587418, 313447090446, 1880711240490, 11284353536382, 67706379498618, 406239051832686, 2437436635519050, 14624626786683102, 87747781640805018

Offset: 0

Henry Bottomley, Apr 05 2001

a(n)/3^n is the expected time to finish a random Tower of Hanoi problem with n disks using optimal moves.

Harry J. Smith, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (9,-18).

Cf. A007798, A060586, A060590.

[2*(2^n - 1)*3^(n - 1): n in [0..30]]; // Vincenzo Librandi, Jul 03 2018

Table[2 (2^n - 1) 3^(n - 1), {n, 0, 50}] (* or *) LinearRecurrence[{9, -18}, {0, 2}, 40] (* Vincenzo Librandi, Jul 03 2018 *)

a(n)={2*(2^n - 1)*3^(n - 1)} \\ Harry J. Smith, Jul 07 2009

a(n) = Sum_{j<2^n} j*A001316(j) = 6*a(n-1) + A008776(n-1) = 4*A000400(n-1) - A008776(n-1) = A000244(n)*A060590(n)/A010684(n).

G.f.: 2*x/((3*x-1)*(6*x-1)). [Colin Barker, Dec 26 2012]

Corrected by T. D. Noe, Nov 07 2006

A060585 Write n in base 3, then (working from left to right) if the k-th digit of n is not equal to the digit to its left then the k-th digit of a(n) is 1, otherwise it is 0, and finally read the result as a base-2 number.

Original entry on oeis.org

0, 1, 1, 3, 2, 3, 3, 3, 2, 6, 7, 7, 5, 4, 5, 7, 7, 6, 6, 7, 7, 7, 6, 7, 5, 5, 4, 12, 13, 13, 15, 14, 15, 15, 15, 14, 10, 11, 11, 9, 8, 9, 11, 11, 10, 14, 15, 15, 15, 14, 15, 13, 13, 12, 12, 13, 13, 15, 14, 15, 15, 15, 14, 14, 15, 15, 13, 12, 13, 15, 15, 14, 10, 11, 11, 11, 10, 11, 9, 9

Offset: 0

Henry Bottomley, Apr 04 2001

A ternary-to-binary map.

Each k appears A001316(k) times in the sequence.

			a(76) = 10 since 76 = 2211_3 and looking for changing digits produces 1010 which, read as if in binary, is 10.

Cf. A001316, A060584, A060586, A060587.

a(n) = my(v=digits(n,3)); if(#v, forstep(k=#v,2,-1, v[k]=(v[k]!=v[k-1])); v[1]=1); fromdigits(v,2); \\ Kevin Ryde, Apr 15 2021

a(n) = 2*a(floor(n/3)) + A060584(n) = A060586(A060587(n)).

Definition rewritten by Editors of OEIS, Apr 15 2021.

cp's OEIS Frontend

A060583 A ternary code related to the Tower of Hanoi.

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A060590 Numerator of the expected time to finish a random Tower of Hanoi problem with n disks using optimal moves.

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A060589 a(n) = 2*(2^n-1)*3^(n-1).

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A060585 Write n in base 3, then (working from left to right) if the k-th digit of n is not equal to the digit to its left then the k-th digit of a(n) is 1, otherwise it is 0, and finally read the result as a base-2 number.

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A060589 a(n) = 2(2^n-1)3^(n-1).