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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.

%I A147646 #33 Feb 24 2021 02:48:18
%S A147646 4,8,12,12,16,28,32,20,16,28,36,40,60,88,80,36,16,28,36,40,60,88,84,
%T A147646 56,60,92,112,140,208,256,192,68,16,28,36,40,60,88,84,56,60,92,112,
%U A147646 140,208,256,196,88,60,92,112,140,208,260,224,172,212,296,364,488,672,704,448,132
%N 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.
%C A147646 Limiting behavior of the rows of the triangle in A139251 when the first column of that triangle is omitted.
%C A147646 First differences of A159795. - _Omar E. Pol_, Jul 24 2009
%C A147646 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 A147646 David Applegate, <a href="/A147646/b147646.txt">Table of n, a(n) for n = 1..2047</a>
%H A147646 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 A147646 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 A147646 <a href="/index/To#toothpick">Index entries for sequences related to toothpick sequences</a>
%F A147646 Letting n = 2^i + j for 0 <= j < 2^i, we have the recurrence (see A139251 for proof):
%F A147646 a(1) = 4
%F A147646 a(2) = 8
%F A147646 a(n) = 2n+4 = 2*a(n/2) - 4 if j = 0
%F A147646 a(n) = 2*a(j) + a(j+1) - 4 if j = 2^i-1
%F A147646 a(n) = 2*a(j) + a(j+1) if 1 <= j < 2^i-1
%F A147646 It appears that a(n) = A151548(n-1) + A151548(n). - _Omar E. Pol_, Feb 19 2015
%e A147646 Further comments: A139251 as a triangle is:
%e A147646 . 1
%e A147646 . 2 4
%e A147646 . 4 4 8 12
%e A147646 . 8 4 8 12 12 16 28 32
%e A147646 . 16 4 8 12 12 16 28 32 20 16 28 36 40 60 88 80
%e A147646 . 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 A147646 leading to the present sequence:
%e A147646 . 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 A147646 Note that this can also be written as a triangle:
%e A147646 . 4 8
%e A147646 . 12 12 16 28
%e A147646 . 32 20 16 28 36 40 60 88
%e A147646 . 80 36 16 28 36 40 60 88 84 56 60 92 112 140 208 256
%e A147646 . 192 68 16 28 36 40 60 88 84 56 60 92 112 140 208 256 196 88 60 92 112 140 ...
%e A147646 The first column is (n+1)2^n (where n is the row number),
%e A147646 the second column is 2^(n+1)+4,
%e A147646 and the rest exhibits the same constant column behavior,
%e A147646 where the rows converge to:
%e A147646 . 16 28 36 40 60 88 84 56 60 92 112 140 208 256 196 88 60 92 112 140 ...
%e A147646 Once again this can be written as a triangle:
%e A147646 . 16
%e A147646 . 28 36 40 60
%e A147646 . 88 84 56 60 92 112 140 208
%e A147646 . 256 196 88 60 92 112 140 208 260 224 172 212 296 364 488 672
%e A147646 . 704 452 152 60 92 112 140 208 260 224 172 212 296 364 488 672 708 480 236 ...
%e A147646 and this behavior continues ad infinitum.
%p A147646 S:=proc(n) option remember; local i,j;
%p A147646 if n <= 0 then RETURN(0); fi;
%p A147646 if n <= 2 then RETURN(2^(n+1)); fi;
%p A147646 i:=floor(log(n)/log(2));
%p A147646 j:=n-2^i;
%p A147646 if j=0 then RETURN(2*n+4); fi;
%p A147646 if j<2^i-1 then RETURN(2*S(j)+S(j+1)); fi;
%p A147646 if j=2^i-1 then RETURN(2*S(j)+S(j+1)-4); fi;
%p A147646 -1;
%p A147646 end; # _N. J. A. Sloane_, May 18 2009
%Y A147646 Equals 2*A151688 and 4*A152980. - _N. J. A. Sloane_, Jul 16 2009
%Y A147646 Cf. A139250, A139251, A153006, A159795, A151548.
%K A147646 nonn
%O A147646 1,1
%A A147646 _David Applegate_, Apr 30 2009