A147646 - OEIS

%I #33 Feb 24 2021 02:48:18

%S 4,8,12,12,16,28,32,20,16,28,36,40,60,88,80,36,16,28,36,40,60,88,84,

%T 56,60,92,112,140,208,256,192,68,16,28,36,40,60,88,84,56,60,92,112,

%U 140,208,256,196,88,60,92,112,140,208,260,224,172,212,296,364,488,672,704,448,132

%N 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.

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

%C First differences of A159795. - _Omar E. Pol_, Jul 24 2009

%C 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

%H David Applegate, <a href="/A147646/b147646.txt">Table of n, a(n) for n = 1..2047</a>

%H David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="/A000695/a000695_1.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>, 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.]

%H N. J. A. Sloane, <a href="/wiki/Catalog_of_Toothpick_and_CA_Sequences_in_OEIS">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>

%H <a href="/index/To#toothpick">Index entries for sequences related to toothpick sequences</a>

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

%F a(1) = 4

%F a(2) = 8

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

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

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

%F It appears that a(n) = A151548(n-1) + A151548(n). - _Omar E. Pol_, Feb 19 2015

%e Further comments: A139251 as a triangle is:

%e . 1

%e . 2 4

%e . 4 4 8 12

%e . 8 4 8 12 12 16 28 32

%e . 16 4 8 12 12 16 28 32 20 16 28 36 40 60 88 80

%e . 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 ...

%e leading to the present sequence:

%e . 4 8 12 12 16 28 32 20 16 28 36 40 60 88 80 36 16 28 36 40 60 88 84 56 ...

%e Note that this can also be written as a triangle:

%e . 4 8

%e . 12 12 16 28

%e . 32 20 16 28 36 40 60 88

%e . 80 36 16 28 36 40 60 88 84 56 60 92 112 140 208 256

%e . 192 68 16 28 36 40 60 88 84 56 60 92 112 140 208 256 196 88 60 92 112 140 ...

%e The first column is (n+1)2^n (where n is the row number),

%e the second column is 2^(n+1)+4,

%e and the rest exhibits the same constant column behavior,

%e where the rows converge to:

%e . 16 28 36 40 60 88 84 56 60 92 112 140 208 256 196 88 60 92 112 140 ...

%e Once again this can be written as a triangle:

%e . 16

%e . 28 36 40 60

%e . 88 84 56 60 92 112 140 208

%e . 256 196 88 60 92 112 140 208 260 224 172 212 296 364 488 672

%e . 704 452 152 60 92 112 140 208 260 224 172 212 296 364 488 672 708 480 236 ...

%e and this behavior continues ad infinitum.

%p S:=proc(n) option remember; local i,j;

%p if n <= 0 then RETURN(0); fi;

%p if n <= 2 then RETURN(2^(n+1)); fi;

%p i:=floor(log(n)/log(2));

%p j:=n-2^i;

%p if j=0 then RETURN(2*n+4); fi;

%p if j<2^i-1 then RETURN(2*S(j)+S(j+1)); fi;

%p if j=2^i-1 then RETURN(2*S(j)+S(j+1)-4); fi;

%p -1;

%p end; # _N. J. A. Sloane_, May 18 2009

%Y Equals 2*A151688 and 4*A152980. - _N. J. A. Sloane_, Jul 16 2009

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

%K nonn

%O 1,1

%A _David Applegate_, Apr 30 2009