OFFSET
1,2
COMMENTS
Bisection of A323651. - Omar E. Pol, Mar 04 2019
REFERENCES
D. Singmaster, On the cellular automaton of Ulam and Warburton, M500 Magazine of the Open University, #195 (December 2003), pp. 2-7.
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
N. J. A. Sloane, Table of n, a(n) for n = 1..10000
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.]
David Applegate, The movie version
Omar E. Pol, Illustration of initial terms (Fig. 1: one-step rook), (Fig. 2: one-step bishop), (Fig. 3: overlapping squares), (Fig. 4: overlapping X-toothpicks), 2009
D. Singmaster, On the cellular automaton of Ulam and Warburton, 2003 [Cached copy, included with permission]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
N. J. A. Sloane, Exciting Number Sequences (video of talk), Mar 05 2021
FORMULA
a(1) = 1; for n > 1, a(n) = 4*3^(wt(n-1)-1) where wt() = A000120(). - R. J. Mathar, Apr 30 2009
This formula is (essentially) given by Singmaster. - N. J. A. Sloane, Aug 06 2009
G.f.: x + 4*x*(Product_{k >= 0} (1 + 3*x^(2^k)) - 1)/3. - N. J. A. Sloane, Jun 10 2009
EXAMPLE
From Omar E. Pol, Jun 14 2009: (Start)
When written as a triangle:
.1;
.4;
.4,12;
.4,12,12,36;
.4,12,12,36,12,36,36,108;
.4,12,12,36,12,36,36,108,12,36,36,108,36,108,108,324;
.4,12,12,36,12,36,36,108,12,36,36,108,36,108,108,324,12,36,36,108,36,108,...
The rows converge to A161411. (End)
MAPLE
A000120 := proc(n) local w, m, i; w := 0; m := n; while m > 0 do i := m mod 2; w := w+i; m := (m-i)/2; od; w; end: wt := A000120; A147582 := n-> if n <= 1 then n else 4*3^(wt(n-1)-1); fi; [seq(A147582(n), n=0..1000)]; # N. J. A. Sloane, Apr 07 2010
MATHEMATICA
s = Plus @@ Flatten@ # & /@ CellularAutomaton[{686, {2, {{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}}, {1, 1}}, {{{1}}, 0}, 200]; f[n_] = If[n == 0, 1, s[[n + 1]] - s[[n]]]; Array[f, 120, 0] (* Michael De Vlieger, Apr 09 2015, after Nadia Heninger and N. J. A. Sloane at A147562 *)
CROSSREFS
Cf. A323651.
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Apr 29 2009
EXTENSIONS
Extended by R. J. Mathar, Apr 30 2009
STATUS
approved