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%I #20 Apr 13 2024 22:16:43
%S 1,2,3,4,5,6,7,65,56,120,56,28,133,30,160,46,24,24,25,28,30,31,31,32,
%T 32,32,35,74,39,38,38,38,39,40,42,319,319,319,319,319,46,129,93,50,50,
%U 72,72,72,72,72,72,53,53,56,56,851,851,167,167,167,167,391
%N The maximum number of alive cells reached in Conway's Game of Life when starting with the first n primes in Ulam's spiral; or -1 if no such maximum exists.
%C The initial alive cells are at coordinates x=A214664(i), y=A214665(i) for i=1..n.
%C For the first 7 terms of this sequence we have a(n)=n since those initial configurations do not lead to complex enough patterns that increase the number of alive cells beyond the initial number of alive cells.
%C The definition includes the possibility that a glider gun (or a similar pattern) is created which will result in an unbounded number of alive cells.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Ulam_spiral">Ulam spiral</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Conway%27s_Game_of_Life">Conway's Game of Life</a>.
%e n=1 to n=4 die out very quickly (within 3 steps). The maximum number of alive cells is simply the number of alive cells in the initial pattern, i.e., n.
%e n=5 is the first term that leads to somewhat interesting steps in the game of life simulation (although the maximum number of alive cells still does not exceed the initial number 5):
%e . . . . . | . . . . . | . . . o . | . . . o . | . . . o . | . . . . .
%e o . o . . | . o o o . | . . o . o | . . o . o | . . . o . | . . . . .
%e . . o o . | . . o o . | . . o . o | . . . . . | . . . . . | . . . . .
%e o . . . . | . . . . . | . . . . . | . . . . . | . . . . . | . . . . .
%e n=8 leads to a maximum number of 65 alive cells and stabilizes after 107 steps. Initial pattern:
%e o . . . o |
%e . o . o . |
%e o . . o o |
%e . o . . . |
%e n=15 reaches a maximum of 160 alive cells and is the first pattern that leads to having a glider (escaping in the northeast direction). Besides the glider, the stabilized pattern contains 4 blinkers, 3 blocks, 2 beehives and 1 ship.
%Y Cf. A214664, A214665.
%Y Cf. A152389, A126803, A099733, A179412.
%K nonn
%O 1,2
%A _Thomas Strohmann_, Mar 01 2024
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