

A160125


Number of squares and rectangles that are created at the nth 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

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), 157191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n1)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(n1)
else
k:=floor(log(n)/log(2)); i:=n2^k;
if (i=0) then RETURN(2^(k1)1)
elif (i<2^k2) then RETURN(2*w(i)+w(i+1));
elif (i=2^k2) 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:=n2^k;
if (i=0) then RETURN(2^k2)
elif (i<=2^k2) 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] (* JeanFrançois 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



