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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

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

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Last modified December 9 23:05 EST 2022. Contains 358710 sequences. (Running on oeis4.)