%I #11 Feb 16 2023 05:12:35
%S 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,
%T 4,13,13,13,40,4,4,13,13,13,40,13,13,40,40,40,121,1,1,4,4,4,13,4,4,13,
%U 13,13,40,4,4,13,13,13,40,13,13,40,40,40,121,4,4,13,13,13,40,13,13,40,40,40
%N a(n) = (3^(wt(k)+f(j))-1)/2 if n = 6k+j, 0 <= j < 6, where wt = A000120, f = A151899.
%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>
%F a(n) = (3^A151902(n)-1)/2.
%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;
%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 a[n_] := (3^A151902[n] - 1)/2;
%t Table[a[n], {n, 0, 82}] (* _Jean-François Alcover_, Feb 16 2023 *)
%o (PARI) a(n)=(3^(hammingweight(n\6)+[0,0,1,1,1,2][n%6+1])-1)/2 \\ _Charles R Greathouse IV_, Sep 26 2015
%Y Cf, A151899, A151902, A151905-A151907.
%K nonn,easy
%O 0,6
%A _N. J. A. Sloane_, Jul 31 2009
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