

A147646


If A139251 is written as a triangle with rows of lengths 1, 2, 4, 8, 16, ..., the nth row begins with 2^n followed by the first 2^n1 terms of the present sequence.


13



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
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OFFSET

1,1


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 nth 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


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), 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
Index entries for sequences related to toothpick sequences


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^i1
a(n) = 2*a(j) + a(j+1) if 1 <= j < 2^i1
It appears that a(n) = A151548(n1) + A151548(n).  Omar E. Pol, Feb 19 2015


EXAMPLE

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.


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:=n2^i;
if j=0 then RETURN(2*n+4); fi;
if j<2^i1 then RETURN(2*S(j)+S(j+1)); fi;
if j=2^i1 then RETURN(2*S(j)+S(j+1)4); fi;
1;
end; # N. J. A. Sloane, May 18 2009


CROSSREFS

Equals 2*A151688 and 4*A152980.  N. J. A. Sloane, Jul 16 2009
Cf. A139250, A139251, A153006, A159795, A151548.
Sequence in context: A120427 A060830 A080458 * A080229 A284510 A167523
Adjacent sequences: A147643 A147644 A147645 * A147647 A147648 A147649


KEYWORD

nonn


AUTHOR

David Applegate, Apr 30 2009


STATUS

approved



