A151906 a(0) = 0, a(1) = 1; for n>1, a(n) = 8*A151905(n) + 4.

0, 1, 4, 4, 4, 12, 4, 4, 12, 12, 12, 36, 4, 4, 12, 12, 12, 36, 12, 12, 36, 36, 36, 108, 4, 4, 12, 12, 12, 36, 12, 12, 36, 36, 36, 108, 12, 12, 36, 36, 36, 108, 36, 36, 108, 108, 108, 324, 4, 4, 12, 12, 12, 36, 12, 12, 36, 36, 36, 108, 12, 12, 36, 36, 36, 108, 36, 36, 108, 108, 108

Offset

0,3

Comments

Consider the Holladay-Ulam CA shown in Fig. 2 and Example 2 of the Ulam article. Then a(n) is the number of cells turned ON in generation n.

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..70.

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 three trisections are essentially A147582, A147582 and 3*A147582 respectively. More precisely, For t >= 1, a(3t) = a(3t+1) = A147582(t+1) = 4*3^(wt(t)-1), a(3t+2) = 4*A147582(t+1) = 4*3^wt(t). See A151907 for explanation.

Example

From Omar E. Pol, Apr 02 2018: (Start)
Note that this sequence also can be written as an irregular triangle read by rows in which the row lengths are the terms of A011782 multiplied by 3, as shown below:
0,1, 4;
4,4,12;
4,4,12,12,12,36;
4,4,12,12,12,36,12,12,36,36,36,108;
4,4,12,12,12,36,12,12,36,36,36,108,12,12,36,36,36,108,36,36,108,108,108,324;
4,4,12,12,12,36,12,12,36,36,36,108,12,12,36,36,36,108,36,36,108,108,108,... (End)

Maple

f := proc(n) local j; j:=n mod 6; if (j<=1) then 0 elif (j<=4) then 1 else 2; fi; end;
wt := 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;
A151904 := proc(n) local k, j; k:=floor(n/6); j:=n-6*k; (3^(wt(k)+f(j))-1)/2; end;
A151905 := proc (n) local k, j;
if (n=0) then 0;
elif (n=1) then 1;
elif (n=2) then 0;
else k:=floor( log(n/3)/log(2) ); j:=n-3*2^k; A151904(j); fi;
end;
A151906 := proc(n);
if (n=0) then 0;
elif (n=1) then 1;
else 8*A151905(n) + 4;
fi;
end;

Mathematica

wt[n_] := DigitCount[n, 2, 1];
f[n_] := {0, 0, 1, 1, 1, 2}[[Mod[n, 6] + 1]];
A151902[n_] := wt[Floor[n/6]] + f[n - 6 Floor[n/6]];
A151904[n_] := (3^A151902[n] - 1)/2;
A151905[n_] := Module[{k, j}, Switch[n, 0, 0, 1, 1, 2, 0, _, k = Floor[Log2[n/3]]; j = n - 3*2^k; A151904[j]]];
a[n_] := Switch[n, 0, 0, 1, 1, _, 8 A151905[n] + 4];
Table[a[n], {n, 0, 70}] (* Jean-François Alcover, Feb 16 2023, after Maple code *)

Crossrefs

Cf. A151904, A151905, A151907, A139250, A151895, A151896.

Sequence in context: A219578 A219373 A337132 * A151896 A267191 A170897

Adjacent sequences: A151903 A151904 A151905 * A151907 A151908 A151909

Keyword

nonn,tabf

Author

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

Status

approved