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A151905
a(0) = a(2) = 0, a(1) = 1; for n >= 3, n = 3*2^k+j, 0 <= j < 3*2^k, a(n) = A151904(j).
6
0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 4, 0, 0, 1, 1, 1, 4, 1, 1, 4, 4, 4, 13, 0, 0, 1, 1, 1, 4, 1, 1, 4, 4, 4, 13, 1, 1, 4, 4, 4, 13, 4, 4, 13, 13, 13, 40, 0, 0, 1, 1, 1, 4, 1, 1, 4, 4, 4, 13, 1, 1, 4, 4, 4, 13, 4, 4, 13, 13, 13, 40, 1, 1, 4, 4, 4, 13, 4, 4, 13, 13, 13, 40, 4, 4, 13, 13, 13, 40, 13
OFFSET
0,12
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 in a 45-degree sector that are not on the main stem.
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, 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, Illustration of initial terms (annotated copy of figure on p. 222 of Ulam)
EXAMPLE
If written as a triangle:
0,
1, 0,
0, 0, 1,
0, 0, 1, 1, 1, 4,
0, 0, 1, 1, 1, 4, 1, 1, 4, 4, 4, 13,
0, 0, 1, 1, 1, 4, 1, 1, 4, 4, 4, 13, 1, 1, 4, 4, 4, 13, 4, 4, 13, 13, 13, 40
0, 0, 1, 1, 1, 4, 1, 1, 4, 4, 4, 13, 1, 1, 4, 4, 4, 13, 4, 4, 13, 13, 13, 40, 1, 1, 4, 4, 4, 13, 4, 4, 13, 13, 13, 40, 4, 4, 13, 13, 13, 40, 13, 13, 40, 40, 40, 121,
...
then the rows converge to A151904.
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;
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;
a[n_] := Module[{k, j}, Switch[n, 0, 0, 1, 1, 2, 0, _, k = Floor[Log2[n/3]]; j = n - 3*2^k; A151904[j]]];
Table[a[n], {n, 0, 90}] (* Jean-François Alcover, Feb 16 2023, after Maple code *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
N. J. A. Sloane, Jul 31 2009
STATUS
approved