A151907 Partial sums of A151906.

0, 1, 5, 9, 13, 25, 29, 33, 45, 57, 69, 105, 109, 113, 125, 137, 149, 185, 197, 209, 245, 281, 317, 425, 429, 433, 445, 457, 469, 505, 517, 529, 565, 601, 637, 745, 757, 769, 805, 841, 877, 985, 1021, 1057, 1165, 1273, 1381, 1705, 1709, 1713, 1725, 1737, 1749, 1785, 1797

Offset: 0

Author

David Applegate and N. J. A. Sloane, Jul 31 2009, Aug 03 2009

Keywords

• nonn

Comments

a(n) is the total number of ON cells after n generations in the Holladay-Ulam CA shown in Fig. 2 and Example 2 of the Ulam article.

The definition of this CA given by Ulam on pp. 216, 222 is complicated (and incomplete). However, the same structure can be obtained as follows. Take the CA of A147562 but replace each square by a Maltese cross, a cluster of five adjacent squares:

..X..

.XXX.

..X..

We guess that this was how Holladay and Ulam originally constructed the CA.

This construction corresponds to the fact that the three trisections of the difference sequence A151906 are essentially A147582, A147582 and 3*A147582 respectively.

References

• S. Ulam, On some mathematical problems connected with patterns of growth of figures, pp. 215-224 of R. E. Bellman, ed., Mathematical Problems in the Biological Sciences, Proc. Sympos. Applied Math., Vol. 14, Amer. Math. Soc., 1962.

Links

• David Applegate, The movie version
• 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.]
• N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
• N. J. A. Sloane, Illustration of initial terms (annotated copy of figure on p. 222 of Ulam)

Crossrefs

Cf. A151904, A151905, A151906, A139250, A151895, A151896.

Formula

The definition in terms of A151906 provides an explicit formula for a(n).