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A160125 Number of squares and rectangles that are created at the n-th stage in the toothpick structure (see A139250). 9
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
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.]
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.
Sequence in context: A009545 A084102 A221609 * A151868 A344913 A052079
KEYWORD
nonn
AUTHOR
Omar E. Pol, May 03 2009
EXTENSIONS
Terms beyond a(10) from R. J. Mathar, Jan 21 2010
STATUS
approved

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Last modified March 29 11:14 EDT 2024. Contains 371278 sequences. (Running on oeis4.)