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A160125 Number of squares and rectangles that are created at the n-th stage in the toothpick structure (see A139250).

%I A160125 #13 Feb 24 2021 02:48:18
%S A160125 0,0,2,2,0,4,10,6,0,4,8,4,4,20,30,14,0,4,8,4,4,20,28,12,4,16,20,12,28,
%T A160125 72,78,30,0,4,8,4,4,20,28,12,4,16,20,12,28,72,76,28,4,16,20,12,28,68,
%U A160125 68,28,24,52,52,52,128,224,190,62,0,4,8,4,4,20,28,12,4,16,20,12,28,72
%N A160125 Number of squares and rectangles that are created at the n-th stage in the toothpick structure (see A139250).
%H A160125 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 A160125 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 A160125 See Maple program for recurrence.
%p A160125 # First construct A168131:
%p A160125 w := proc(n) option remember; local k,i;
%p A160125 if (n=0) then RETURN(0)
%p A160125 elif (n <= 3) then RETURN(n-1)
%p A160125 else
%p A160125 k:=floor(log(n)/log(2)); i:=n-2^k;
%p A160125 if (i=0) then RETURN(2^(k-1)-1)
%p A160125 elif (i<2^k-2) then RETURN(2*w(i)+w(i+1));
%p A160125 elif (i=2^k-2) then RETURN(2*w(i)+w(i+1)+1);
%p A160125 else RETURN(2*w(i)+w(i+1)+2);
%p A160125 fi; fi; end;
%p A160125 # Then construct A160125:
%p A160125 r := proc(n) option remember; local k,i;
%p A160125 if (n<=2) then RETURN(0)
%p A160125 elif (n <= 4) then RETURN(2)
%p A160125 else
%p A160125 k:=floor(log(n)/log(2)); i:=n-2^k;
%p A160125 if (i=0) then RETURN(2^k-2)
%p A160125 elif (i<=2^k-2) then RETURN(4*w(i));
%p A160125 else RETURN(4*w(i)+2);
%p A160125 fi; fi; end;
%p A160125 [seq(r(n),n=0..200)];
%p A160125 # _N. J. A. Sloane_, Feb 01 2010
%t A160125 w [n_] := w[n] = Module[{k, i}, Which[n == 0, 0, n <= 3, n - 1, True, k = Floor[Log[2, n]]; i = n - 2^k; Which[i == 0, 2^(k - 1) - 1, i < 2^k - 2, 2 w[i] + w[i + 1], i == 2^k - 2, 2 w[i] + w[i + 1] + 1, True, 2 w[i] + w[i + 1] + 2]]];
%t A160125 r[n_] := r[n] = Module[{k, i}, Which[n <= 2, 0, n <= 4, 2, True, k = Floor[Log[2, n]]; i = n - 2^k; Which[i == 0, 2^k - 2, i <= 2^k - 2, 4 w[i], True, 4 w[i] + 2]]];
%t A160125 Array[r, 78] (* _Jean-François Alcover_, Apr 15 2020, from Maple *)
%Y A160125 First differences of A160124.
%Y A160125 Cf. toothpick sequence A139250.
%Y A160125 Cf. A159786, A159787, A159788, A159789, A160124, A160126, A160127.
%K A160125 nonn
%O A160125 1,3
%A A160125 _Omar E. Pol_, May 03 2009
%E A160125 Terms beyond a(10) from _R. J. Mathar_, Jan 21 2010