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

1,3

Links

Table of n, a(n) for n=1..78.

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

Formula

See Maple program for recurrence.

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

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 *)

Crossrefs

First differences of A160124.

Cf. toothpick sequence A139250.

Cf. A159786, A159787, A159788, A159789, A160124, A160126, A160127.

Sequence in context: A009545 A084102 A221609 * A151868 A344913 A052079

Adjacent sequences: A160122 A160123 A160124 * A160126 A160127 A160128

Keyword

nonn

Author

Omar E. Pol, May 03 2009

Extensions

Terms beyond a(10) from R. J. Mathar, Jan 21 2010

Status

approved