A268284 Period 15: repeat {18, 20, 28, 20, 20, 22, 18, 22, 20, 16, 12, 22, 18, 40, 18}.

18, 20, 28, 20, 20, 22, 18, 22, 20, 16, 12, 22, 18, 40, 18, 18, 20, 28, 20, 20, 22, 18, 22, 20, 16, 12, 22, 18, 40, 18, 18, 20, 28, 20, 20, 22, 18, 22, 20, 16, 12, 22, 18, 40, 18, 18, 20, 28, 20, 20, 22, 18, 22, 20, 16, 12, 22, 18, 40, 18, 18, 20, 28, 20, 20, 22, 18, 22, 20, 16, 12, 22, 18, 40, 18

Offset: 0

Author

Ilya Gutkovskiy, Jan 30 2016

Keywords

• nonn
• easy

Comments

Number of living cells periodic figure (oscillators: pentadecathlon (period 15)) in the Conway's Game of Life (rule B3/S23: see Graphical example in Links section).

Examples

			Start pattern (see Graphical example in Links section):
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|. . . . . . . . . . . . . . . .| . . . . . . . . . . . . . . . .|
|. . . . o . . . . . . o . . . .| . . . o o . . . . . . o o . . .|
|. . . o o . . . . . . o o . . .| . . o . . o . . . . o . . o . .|
|. . o o o . . . . . . o o o . .| . . 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), etc.            |
		

Links

• Golly, Cross-platform application for exploring Conway's Game of Life and other cellular automata
• Ilya Gutkovskiy, Graphical example (generation 0-15)
• Eric Weisstein's World of Mathematics, Game of Life
• Wikipedia, Conway's Game of Life
• Wikipedia, Oscillator (cellular automaton)
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,0,1).

Crossrefs

Cf. A061342, A152389, A179409.

Programs

Magma

&cat[[18,20,28,20,20,22,18,22,20,16,12,22,18,40,18]^^7]; // Vincenzo Librandi, Jan 30 2016

Mathematica

LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1}, {18, 20, 28, 20, 20, 22, 18, 22, 20, 16, 12, 22, 18, 40, 18}, 80]

PARI

a(n)=2*[9, 10, 14, 10, 10, 11, 9, 11, 10, 8, 6, 11, 9, 20, 9][n%15+1] \\ Charles R Greathouse IV, Jul 17 2016

Formula

For k>=0:

a((30*k - 2*sin((Pi*k)/2) - 18*cos((Pi*k)/2) - cos(Pi*k) + 19)/8) = 18;

a((30*k + 10*sin((Pi*k)/2) + 18*cos((Pi*k)/2) + 3*cos(Pi*k) - 13)/8) = 20;

a(15*k + 2) = 28;

a(15*k + 9) = 16;

a(15*k + 10) = 12;

a(15*k + 13) = 40.