A137605 -id:A137605 - cp's OEIS frontend

A096259 Longest period of an abstract version of the game of Go on a 1 X n board.

1, 2, 6, 24, 70, 180, 294, 112, 270, 900, 330, 792

Offset: 1

Yasutoshi Kohmoto, Aug 01 2004; revised Apr 23 2008

Rules: 1. If a set of a player's stones has no "open edge" then the other player get the set of stones.
2. If the sets of both player's stones has no "open edge" in a configuration, then a player who made this configuration get the set of the other player's stone.
3. A player never make a configuration in which his stones have no open edge and the other player's stones have an open edge.
A board is represented as follows.
  + + + +
  + o x +
  + + + +
"o" means a white stone, "x" means a black stone.
"Open edge" : An edge which has one node without a stone. Example:
  + x x +
  x o o x
  + x x +
The center set of white stones has no "open edge", so black player gets them. Six black stones have "open edges" like this : "x +".
Note that the rules do not specify when a player wins, so the game never terminates.

			The case n=3:
  t 1 2 3 3 4 4 5 6 6 7 7
  + x x x x x + x x + x x
  + + + x x x + + o o o +
  + + o o + o o o o o o +
t=1 and t=7 are the same, so the period is 6.
a(12) = 12 * 2^0 * (12 + 6 + 3 + 10 + 5 + 9 + 7 + 8 + 4 + 2 + 1 - 1) = 792.

Cf. A007565, A048289, A137604, A137605, A137606, A137607.

For 4<=n, a(n) = n * 2^p * ( Sum_{0<=k<=m} ( Sum_{0<=i<=h_k} n_k/2^i ) - 1 ) where p = m Mod 2, n_0 = n, n_k = n - [n_{k-1}/2^(h_{k-1}+1)] - 1, 2^h_k is the highest power of two dividing n_k: n_m/2^h_m = 1.

A137604 Consider the sequence: b(0) = n, and for k >= 1, b(k) = b(k-1)/2 if b(k-1) is even, otherwise b(k) = n - (b(k-1)+1)/2. Then a(n) = (Sum_{0<=k<=m} b(k)) - 1 for n > 1, where m is the smallest index such that b(m) = 1; a(1) = 1.

Original entry on oeis.org

1, 2, 3, 6, 7, 15, 21, 14, 15, 45, 30, 66, 63, 61, 105, 30, 31, 102, 171, 114, 93, 63, 134, 276, 258, 88, 351, 270, 105, 435, 465, 62, 63, 561, 374, 630, 126, 374, 570, 780, 547, 861, 126, 602, 204, 246, 196, 846, 537, 361, 1275, 1326, 264, 1431, 483, 990, 315

Offset: 1

Yasutoshi Kohmoto, Apr 23 2008

nonn

			a(6) = 6 + 3 + 4 + 2 + 1 - 1 = 15.

Cf. A096259, A137605.

f[1] = 1; f[n_] := Block[{lst = {n}, a}, While[a = lst[[ -1]]; a != 1, If[EvenQ@ a, AppendTo[lst, a/2], AppendTo[lst, lst[[1]] - (a + 1)/2]]]; Plus @@ lst - 1]; Array[f, 58] (* Robert G. Wilson v, May 15 2008 *)

More terms from Robert G. Wilson v, May 15 2008

A139180 Smallest number m greater than 1 such that m=(Sum_{i=1..k} d_i!)^n where d_1 d_2 ... d_k is the decimal expansion of m, or 0 if no m exists.

Original entry on oeis.org

2, 172603687936, 3910825640696988096, 396867211295273131775761, 13586030059671762261567625351, 4218282992523473247642107344280879104, 5572272365289149102816175194628254483025277

Offset: 1

Farideh Firoozbakht, Apr 22 2008

A139181(n)=A139180(n)^(1/n).

			172603687936=(1!+7!+2!+6!+0!+3!+6!+8!+7!+9!+3!+6!)^2

Cf. A014080, A137604, A137605.

cp's OEIS Frontend

A096259 Longest period of an abstract version of the game of Go on a 1 X n board.

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A137604 Consider the sequence: b(0) = n, and for k >= 1, b(k) = b(k-1)/2 if b(k-1) is even, otherwise b(k) = n - (b(k-1)+1)/2. Then a(n) = (Sum_{0<=k<=m} b(k)) - 1 for n > 1, where m is the smallest index such that b(m) = 1; a(1) = 1.

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A139180 Smallest number m greater than 1 such that m=(Sum_{i=1..k} d_i!)^n where d_1 d_2 ... d_k is the decimal expansion of m, or 0 if no m exists.

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