A147582 First differences of A147562.

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, 108, 324, 36, 108, 108, 324, 108, 324, 324, 972, 4

Offset: 1

Author

N. J. A. Sloane, Apr 29 2009

Keywords

• nonn

Comments

Bisection of A323651. - Omar E. Pol, Mar 04 2019

Examples

			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)
		

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
• Omar E. Pol, Illustration of initial terms of A139251, A160121, A147582 (Overlapping figures), 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

Crossrefs

Cf. A147562, A147610 (the sequence divided by 4), A048881, A000120.

Cf. A000079, A161411, A151779, A139250.

Cf. A048883, A139251, A160121, A162349. [Omar E. Pol, Nov 02 2009]

Cf. A323651.

Programs

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 *)

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

Extensions

Extended by R. J. Mathar, Apr 30 2009