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%I #12 Feb 16 2023 05:12:43
%S 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,
%T 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,
%U 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
%N 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).
%C 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.
%D 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.
%H David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="/A000695/a000695_1.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>, 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.]
%H N. J. A. Sloane, <a href="/wiki/Catalog_of_Toothpick_and_CA_Sequences_in_OEIS">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%H N. J. A. Sloane, <a href="/A151907/a151907.jpg">Illustration of initial terms</a> (annotated copy of figure on p. 222 of Ulam)
%e If written as a triangle:
%e 0,
%e 1, 0,
%e 0, 0, 1,
%e 0, 0, 1, 1, 1, 4,
%e 0, 0, 1, 1, 1, 4, 1, 1, 4, 4, 4, 13,
%e 0, 0, 1, 1, 1, 4, 1, 1, 4, 4, 4, 13, 1, 1, 4, 4, 4, 13, 4, 4, 13, 13, 13, 40
%e 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,
%e ...
%e then the rows converge to A151904.
%p f := proc(n) local j; j:=n mod 6; if (j<=1) then 0 elif (j<=4) then 1 else 2; fi; end;
%p 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;
%p A151904 := proc(n) local k,j; k:=floor(n/6); j:=n-6*k; (3^(wt(k)+f(j))-1)/2; end;
%p A151905 := proc (n) local k,j;
%p if (n=0) then 0;
%p elif (n=1) then 1;
%p elif (n=2) then 0;
%p else k:=floor( log(n/3)/log(2) ); j:=n-3*2^k; A151904(j); fi;
%p end;
%t wt[n_] := DigitCount[n, 2, 1];
%t f[n_] := {0, 0, 1, 1, 1, 2}[[Mod[n, 6] + 1]];
%t A151902[n_] := wt[Floor[n/6]] + f[n - 6 Floor[n/6]];
%t A151904[n_] := (3^A151902[n] - 1)/2;
%t 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]]];
%t Table[a[n], {n, 0, 90}] (* _Jean-François Alcover_, Feb 16 2023, after Maple code *)
%Y Cf. A151904, A151906, A151907, A139250, A151895, A151896.
%K nonn,tabf
%O 0,12
%A _N. J. A. Sloane_, Jul 31 2009
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