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%I #21 Sep 25 2022 11:28:08
%S 0,0,1,2,1,1,5,7,3,1,4,5,3,7,18,19,7,1,4,5,3,7,17,17,7,6,13,13,13,32,
%T 56,47,15,1,4,5,3,7,17,17,7,6,13,13,13,32,55,45,15,6,13,13,13,31,51,
%U 41,20,25,39,39,58,120,160,111,31,1,4,5,3,7,17,17,7,6,13,13,13,32,55,45,15,6
%N Number of squares and rectangles that are created at the n-th stage in the corner toothpick structure (see A152980, A153006).
%C Essentially the first differences of A170926. - _Omar E. Pol_, Feb 16 2013
%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 See Maple program for recurrence.
%e If written as a triangle:
%e 0,
%e 0,
%e 1,2,
%e 1,1,5,7,
%e 3,1,4,5,3,7,18,19,
%e 7,1,4,5,3,7,17,17,7,6,13,13,13,32,56,47,
%e 15,1,4,5,3,7,17,17,7,6,13,13,13,32,55,45,15,6,13,13,13,31,51,41,20,...
%e The rows (omitting the first term) converge to A170929.
%p w := proc(n) option remember; local k,i;
%p if (n=0) then RETURN(0)
%p elif (n <= 3) then RETURN(n-1)
%p else
%p k:=floor(log(n)/log(2));
%p i:=n-2^k;
%p if (i=0) then RETURN(2^(k-1)-1)
%p elif (i<2^k-2) then RETURN(2*w(i)+w(i+1));
%p elif (i=2^k-2) then RETURN(2*w(i)+w(i+1)+1);
%p else RETURN(2*w(i)+w(i+1)+2);
%p fi;
%p fi;
%p end;
%p [seq(w(n),n=0..256)];
%t a[n_] := a[n] = Module[{k, i}, Which[n==0, 0, n <= 3, n-1, True, k = Floor[Log2[n]]; i = n-2^k; Which[i==0, 2^(k-1)-1, i < 2^k-2, 2*a[i]+a[i+1], i==2^k-2, 2*a[i]+a[i+1]+1, True, 2*a[i]+a[i+1]+2]]];
%t Table[a[n], {n, 0, 81}] (* _Jean-François Alcover_, Sep 25 2022, after Maple code *)
%Y Cf. A152980, A153006, A170926, A160124, A160125, A139250.
%K nonn
%O 0,4
%A _Omar E. Pol_, Jan 18 2010
%E Edited and extended by _N. J. A. Sloane_, Feb 01 2010
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