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

0,3

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

Table of n, a(n) for n=0..54.

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)

Formula

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

Crossrefs

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

Sequence in context: A208718 A208774 A271391 * A151895 A267190 A170896

Adjacent sequences: A151904 A151905 A151906 * A151908 A151909 A151910

Keyword

nonn

Author

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

Status

approved