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A170927
Consider the 2^n values of A139250(i)/i^2 for 2^n <= i < 2^(n+1); a(n) = value of i where this quantity is minimized.
6
1, 2, 5, 12, 21, 44, 89, 180, 362, 728, 1459, 2921, 5843, 11690, 23384, 46770, 93544, 187094, 374193, 748391, 1496786, 2993576, 5987158, 11974321, 23948647, 47897300, 95794608, 191589222, 383178450, 766356910, 1532713828, 3065427664, 6130855333, 12261710675
OFFSET
0,2
COMMENTS
{log_2 a(n)} converges to about 0.513441 and equivalently 2^{log_2 a(n)}-1 converges to about 1.427451, and the corresponding values T(i)/i^2 converge to about 0.4513058.
For all values listed, a(n) = 2 * a(n-1) + c(n), where c(n) is a small positive integer, except for a(4) where c(4)=-3. - Robert Price, Aug 16 2015
LINKS
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
Steven R. Finch, Toothpicks and Live Cells, July 21, 2015. [Cached copy, with permission of the author]
EXAMPLE
The values of A139250(i)/i^2 for i = 1 .., 15 are 1.0, 0.7500000000, 0.7777777778, 0.6875000000, 0.6000000000, 0.6388888889, 0.7142857143, 0.6718750000, 0.5802469136, 0.5500000000, 0.5537190083, 0.5486111111, 0.5621301775, 0.6275510204, 0.6888888889, 0.6679687500. The minimal value for 4 <= i <= 7 is 0.6000000000 at i=5.
CROSSREFS
KEYWORD
nonn
AUTHOR
Benoit Jubin, Jan 22 2010, Feb 06 2010
EXTENSIONS
a(26)-a(33) from Robert Price, Aug 18 2012
STATUS
approved