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.

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

Author

Benoit Jubin, Jan 22 2010, Feb 06 2010

Keywords

• nonn

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

Examples

			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.
		

Links

• Robert Price, Table of n, a(n) for n = 0..169
• 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]
• N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Crossrefs

Cf. A139250, A170927, A147562, A260239, A261313.

Extensions

a(26)-a(33) from Robert Price, Aug 18 2012