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

Author

Omar E. Pol, Jan 18 2010

Keywords

• nonn

Comments

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

Examples

			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.
		

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.]
• N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Crossrefs

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

Programs

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 *)

Formula

See Maple program for recurrence.

Extensions

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