This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A151906 #21 Feb 16 2023 05:12:49
%S A151906 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,
%T A151906 12,12,12,36,12,12,36,36,36,108,12,12,36,36,36,108,36,36,108,108,108,
%U A151906 324,4,4,12,12,12,36,12,12,36,36,36,108,12,12,36,36,36,108,36,36,108,108,108
%N A151906 a(0) = 0, a(1) = 1; for n>1, a(n) = 8*A151905(n) + 4.
%C A151906 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.
%D A151906 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 A151906 David Applegate, <a href="/A139250/a139250.anim.html">The movie version</a>
%H A151906 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 A151906 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 A151906 N. J. A. Sloane, <a href="/A151907/a151907.jpg">Illustration of initial terms</a> (annotated copy of figure on p. 222 of Ulam)
%F A151906 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.
%e A151906 From _Omar E. Pol_, Apr 02 2018: (Start)
%e A151906 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:
%e A151906 0,1, 4;
%e A151906 4,4,12;
%e A151906 4,4,12,12,12,36;
%e A151906 4,4,12,12,12,36,12,12,36,36,36,108;
%e A151906 4,4,12,12,12,36,12,12,36,36,36,108,12,12,36,36,36,108,36,36,108,108,108,324;
%e A151906 4,4,12,12,12,36,12,12,36,36,36,108,12,12,36,36,36,108,36,36,108,108,108,... (End)
%p A151906 f := proc(n) local j; j:=n mod 6; if (j<=1) then 0 elif (j<=4) then 1 else 2; fi; end;
%p A151906 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 A151906 A151904 := proc(n) local k,j; k:=floor(n/6); j:=n-6*k; (3^(wt(k)+f(j))-1)/2; end;
%p A151906 A151905 := proc (n) local k,j;
%p A151906 if (n=0) then 0;
%p A151906 elif (n=1) then 1;
%p A151906 elif (n=2) then 0;
%p A151906 else k:=floor( log(n/3)/log(2) ); j:=n-3*2^k; A151904(j); fi;
%p A151906 end;
%p A151906 A151906 := proc(n);
%p A151906 if (n=0) then 0;
%p A151906 elif (n=1) then 1;
%p A151906 else 8*A151905(n) + 4;
%p A151906 fi;
%p A151906 end;
%t A151906 wt[n_] := DigitCount[n, 2, 1];
%t A151906 f[n_] := {0, 0, 1, 1, 1, 2}[[Mod[n, 6] + 1]];
%t A151906 A151902[n_] := wt[Floor[n/6]] + f[n - 6 Floor[n/6]];
%t A151906 A151904[n_] := (3^A151902[n] - 1)/2;
%t A151906 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]]];
%t A151906 a[n_] := Switch[n, 0, 0, 1, 1, _, 8 A151905[n] + 4];
%t A151906 Table[a[n], {n, 0, 70}] (* _Jean-François Alcover_, Feb 16 2023, after Maple code *)
%Y A151906 Cf. A151904, A151905, A151907, A139250, A151895, A151896.
%K A151906 nonn,tabf
%O A151906 0,3
%A A151906 _David Applegate_ and _N. J. A. Sloane_, Jul 31 2009, Aug 03 2009