A147646 If A139251 is written as a triangle with rows of lengths 1, 2, 4, 8, 16, ..., the n-th row begins with 2^n followed by the first 2^n-1 terms of the present sequence.

4, 8, 12, 12, 16, 28, 32, 20, 16, 28, 36, 40, 60, 88, 80, 36, 16, 28, 36, 40, 60, 88, 84, 56, 60, 92, 112, 140, 208, 256, 192, 68, 16, 28, 36, 40, 60, 88, 84, 56, 60, 92, 112, 140, 208, 256, 196, 88, 60, 92, 112, 140, 208, 260, 224, 172, 212, 296, 364, 488, 672, 704, 448, 132

Offset: 1

Author

David Applegate, Apr 30 2009

Keywords

• nonn

Comments

Limiting behavior of the rows of the triangle in A139251 when the first column of that triangle is omitted.

First differences of A159795. - Omar E. Pol, Jul 24 2009

It appears that a(n) is also the number of new grid points that are covered at n-th stage of A139250 version "Tree", assuming the toothpicks have length 4, 3, and 2 (see also A159795 and A153006). - Omar E. Pol, Oct 25 2011

Examples

			Further comments: A139251 as a triangle is:
. 1
. 2 4
. 4 4 8 12
. 8 4 8 12 12 16 28 32
. 16 4 8 12 12 16 28 32 20 16 28 36 40 60 88 80
. 32 4 8 12 12 16 28 32 20 16 28 36 40 60 88 80 36 16 28 36 40 60 88 84 56 ...
leading to the present sequence:
. 4 8 12 12 16 28 32 20 16 28 36 40 60 88 80 36 16 28 36 40 60 88 84 56 ...
Note that this can also be written as a triangle:
. 4 8
. 12 12 16 28
. 32 20 16 28 36 40 60 88
. 80 36 16 28 36 40 60 88 84 56 60 92 112 140 208 256
. 192 68 16 28 36 40 60 88 84 56 60 92 112 140 208 256 196 88 60 92 112 140 ...
The first column is (n+1)2^n (where n is the row number),
the second column is 2^(n+1)+4,
and the rest exhibits the same constant column behavior,
where the rows converge to:
. 16 28 36 40 60 88 84 56 60 92 112 140 208 256 196 88 60 92 112 140 ...
Once again this can be written as a triangle:
. 16
. 28 36 40 60
. 88 84 56 60 92 112 140 208
. 256 196 88 60 92 112 140 208 260 224 172 212 296 364 488 672
. 704 452 152 60 92 112 140 208 260 224 172 212 296 364 488 672 708 480 236 ...
and this behavior continues ad infinitum.
		

Links

• David Applegate, Table of n, a(n) for n = 1..2047
• 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
• Index entries for sequences related to toothpick sequences

Crossrefs

Equals 2*A151688 and 4*A152980. - N. J. A. Sloane, Jul 16 2009

Cf. A139250, A139251, A153006, A159795, A151548.

Programs

Maple

S:=proc(n) option remember; local i,j;
if n <= 0 then RETURN(0); fi;
if n <= 2 then RETURN(2^(n+1)); fi;
i:=floor(log(n)/log(2));
j:=n-2^i;
if j=0 then RETURN(2*n+4); fi;
if j<2^i-1 then RETURN(2*S(j)+S(j+1)); fi;
if j=2^i-1 then RETURN(2*S(j)+S(j+1)-4); fi;
-1;
end; # N. J. A. Sloane, May 18 2009

Formula

Letting n = 2^i + j for 0 <= j < 2^i, we have the recurrence (see A139251 for proof):

a(1) = 4

a(2) = 8

a(n) = 2n+4 = 2*a(n/2) - 4 if j = 0

a(n) = 2*a(j) + a(j+1) - 4 if j = 2^i-1

a(n) = 2*a(j) + a(j+1) if 1 <= j < 2^i-1

It appears that a(n) = A151548(n-1) + A151548(n). - Omar E. Pol, Feb 19 2015