A168131 Number of squares and rectangles that are created at the n-th stage in the corner toothpick structure (see A152980, A153006).

0, 0, 1, 2, 1, 1, 5, 7, 3, 1, 4, 5, 3, 7, 18, 19, 7, 1, 4, 5, 3, 7, 17, 17, 7, 6, 13, 13, 13, 32, 56, 47, 15, 1, 4, 5, 3, 7, 17, 17, 7, 6, 13, 13, 13, 32, 55, 45, 15, 6, 13, 13, 13, 31, 51, 41, 20, 25, 39, 39, 58, 120, 160, 111, 31, 1, 4, 5, 3, 7, 17, 17, 7, 6, 13, 13, 13, 32, 55, 45, 15, 6

Offset

0,4

Comments

Essentially the first differences of A170926. - Omar E. Pol, Feb 16 2013

Links

Table of n, a(n) for n=0..81.

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.

Example

If written as a triangle:
0,
0,
1,2,
1,1,5,7,
3,1,4,5,3,7,18,19,
7,1,4,5,3,7,17,17,7,6,13,13,13,32,56,47,
15,1,4,5,3,7,17,17,7,6,13,13,13,32,55,45,15,6,13,13,13,31,51,41,20,...
The rows (omitting the first term) converge to A170929.

Maple

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;
[seq(w(n), n=0..256)];

Mathematica

a[n_] := a[n] = Module[{k, i}, Which[n==0, 0, n <= 3, n-1, True, k = Floor[Log2[n]]; i = n-2^k; Which[i==0, 2^(k-1)-1, i < 2^k-2, 2*a[i]+a[i+1], i==2^k-2, 2*a[i]+a[i+1]+1, True, 2*a[i]+a[i+1]+2]]];
Table[a[n], {n, 0, 81}] (* Jean-François Alcover, Sep 25 2022, after Maple code *)

Crossrefs

Cf. A152980, A153006, A170926, A160124, A160125, A139250.

Sequence in context: A306444 A090210 A248925 * A024462 A049252 A098315

Adjacent sequences: A168128 A168129 A168130 * A168132 A168133 A168134

Keyword

nonn

Author

Omar E. Pol, Jan 18 2010

Extensions

Edited and extended by N. J. A. Sloane, Feb 01 2010

Status

approved