[M2] (golly 2.0)
#C Patterns whose population grows logarithmically have
#C long been known, but this is believed to be the first pattern
#C whose *diameter* grows logarithmically. It is based on a
#C blinker puffer that can be deleted by its own output (a blinker),
#C or push a block out by 8 cells.
#C
#C By Jason Summers with Gabriel Nivasch, 16 Aug 03
#C
#C Analysis by Dean Hickerson:
#C
#C Let g(n) be the generation in which the leading puffer is in the
#C same position as in gen 0, the block is 8n units to the right of
#C its initial position, and there are no blinkers in between.  I.e.,
#C the puffer is on its way to push the block from position n to
#C position n+1.  g(n) is always a multiple of 240; the first few
#C values are:
#C       n   0   1   2    3    4    5    6    7     8     9    10
#C    g(n)   0 240 720 1440 2400 3840 6000 9120 13680 20400 30240
#C g(n)/240  0   1   3    6   10   16   25   38    57    85   126
#C
#C Letting h(n)=g(n)/240, we have the recurrence
#C    h(n) = h(n-1) + h(n-3) + 3.
#C
#C Also,
#C    h(n) = round(C * R^n) - 3,
#C
#C where
#C   R = (2 + cbrt(116 + 12 sqrt(93)) + cbrt(116 - 12 sqrt(93))) / 6
#C     ~ 1.465571231876768026656731225219939108025577568472285701643
#C
#C and
#C   C = 1 + cbrt((13113 + 151 sqrt(93))/17298) +
#C                                 cbrt((13113 - 151 sqrt(93))/17298)
#C    ~ 2.82110012372929333835936935875070702572316547898891137253679
#C (C satisfies  31 C^3 - 93 C^2 + 16C - 1 = 0.)
#C
#C Since the diameter in generation g(n) is  8n + O(1), we find that
#C the diameter in gen t is asymptotic to  A * log(t),  where
#C    A = 8/log(R)
#C      ~ 20.92898063664810844728999060141708775230924963625537466402
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