A179412 The number of alive cells in Conway's Game of Life on the 8 X 8 toroidal grid, in a cyclic sequence of 132 patterns, whose initial pattern is given in illustrations below.

8, 8, 9, 10, 12, 16, 13, 23, 16, 22, 18, 24, 16, 20, 21, 23, 28, 19, 18, 18, 23, 22, 18, 27, 16, 20, 10, 10, 10, 13, 15, 19, 22, 18, 25, 18, 19, 23, 23, 20, 21, 22, 30, 19, 22, 21, 20, 28, 19, 16, 14, 9, 13, 12, 13, 14, 16, 23, 15, 19, 16, 26, 16, 12, 12, 9, 8, 8, 9, 10, 12, 16

Offset: 0

Author

Antti Karttunen, Jul 27 2010

Keywords

• nonn

Comments

Period 66. The sequence begins (from offset 0) with its lexicographically earliest rotation. Note the almost symmetric subsequence around the terms 66k and 66k+1: ...,16,12,12,9,8,8,9,10,12,16,... All integers in range [8,30] occur except 11, 17 and 29. The mean value of terms in the whole period of 66 is 17.7273.

This is the longest cyclic sequence that I have found so far (July 2010) on 8 X 8 toroidal grid, after the cycle of 48 given in A179409. Are there any longer cyclic sequences? A sequence to be computed: for n X n toroidal grid, the longest cycle of patterns that can occur. (Also other metrics for toroidal boards: how many patterns die in next generation, how many are stable, etc.)

Examples

			The generations 0-3 of this cycle of patterns look as follows, thus a(0)=a(1)=8, a(2)=9 and a(3)=10. Note how the initial pattern differs by just one misplaced cell from the pattern present in the generation 3 of A179409.
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. o o . . . . . | . o o . . . . . | . o o . . . . . | o o o . . . . .
. o . o . . . . | . o . o . . . . | o o . o . . . . | o . . o . . . .
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(generation 0.) | (generation 1.) | (generation 2.) | (generation 3.)
In generation 66 we obtain a mirror image of the initial pattern, and in the generations 66--131 the patterns repeat the history of the first 66 generations, but reflected over the vertical axis, after which the whole cycle begins from the start again, at the generation 132.
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(generation 66) | (generation 67)
		

Links

• Antti Karttunen, J-program for computing the terms of this sequence
• Antti Karttunen, Electronic music project using this Life-pattern sequence as one of its preferred sources of variation
• Index entries for sequences related to music

Crossrefs

Cf. A179413, A179414, A179409.

Cf. also A060118 (another input for the music project).