A160125 Number of squares and rectangles that are created at the n-th stage in the toothpick structure (see A139250).
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, 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, 68, 28, 24, 52, 52, 52, 128, 224, 190, 62, 0, 4, 8, 4, 4, 20, 28, 12, 4, 16, 20, 12, 28, 72
Offset: 1
Keywords
Links
- David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, 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.]
- N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Crossrefs
Programs
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Maple
# First construct A168131: w := proc(n) option remember; local k,i; if (n=0) then RETURN(0) elif (n <= 3) then RETURN(n-1) else k:=floor(log(n)/log(2)); i:=n-2^k; if (i=0) then RETURN(2^(k-1)-1) elif (i<2^k-2) then RETURN(2*w(i)+w(i+1)); elif (i=2^k-2) then RETURN(2*w(i)+w(i+1)+1); else RETURN(2*w(i)+w(i+1)+2); fi; fi; end; # Then construct A160125: r := proc(n) option remember; local k,i; if (n<=2) then RETURN(0) elif (n <= 4) then RETURN(2) else k:=floor(log(n)/log(2)); i:=n-2^k; if (i=0) then RETURN(2^k-2) elif (i<=2^k-2) then RETURN(4*w(i)); else RETURN(4*w(i)+2); fi; fi; end; [seq(r(n),n=0..200)]; # N. J. A. Sloane, Feb 01 2010
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Mathematica
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]]]; 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]]]; Array[r, 78] (* Jean-François Alcover, Apr 15 2020, from Maple *)
Formula
See Maple program for recurrence.
Extensions
Terms beyond a(10) from R. J. Mathar, Jan 21 2010