A194443 - OEIS

%I #22 Sep 26 2015 01:17:13

%S 0,1,2,4,4,4,4,7,8,4,4,8,12,8,8,13,16,4,4,8,12,16,16,20,24,12,8,16,28,

%T 16,16,25,32,4,4,8,12,16,16,22,32,26,20,24,40,32,40,33,48,20,8,16,28,

%U 40,44,50,60,28,16,32,60,32,32,49,64,4,4,8

%N Number of toothpicks or D-toothpicks added at n-th stage to the structure of A194442.

%C Essentially the first differences of A194442. It appears that the structure of the "narrow" triangle is much more regular about n=2^k, see formula section.

%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 Conjectures for n = 2^k+j, if -6<=j<=6:

%F a(2^k-6) = 2^(k-2), if k >= 3.

%F a(2^k-5) = 2^(k-1), if k >= 3.

%F a(2^k-4) = 2^k-4, if k >= 2.

%F a(2^k-3) = 2^(k-1), if k >= 3.

%F a(2^k-2) = 2^(k-1), if k >= 2.

%F a(2^k-1) = 3*2^(k-2)+1, if k >= 2.

%F a(2^k+0) = 2^k, if k >= 0.

%F a(2^k+1) = 4, if k >= 1.

%F a(2^k+2) = 4, if k >= 1.

%F a(2^k+3) = 8, if k >= 3.

%F a(2^k+4) = 12, if k >= 3.

%F a(2^k+5) = 16, if k >= 4.

%F a(2^k+6) = 16, if k >= 4.

%F End of conjectures.

%e If written as a triangle:

%e 0,

%e 1,

%e 2,

%e 4,4,

%e 4,4,7,8,

%e 4,4,8,12,8,8,13,16,

%e 4,4,8,12,16,16,20,24,12,8,16,28,16,16,25,32,

%e 4,4,8,12,16,16,22,32,26,20,24,40,32,40,33,48,20,8,16,28...

%e .

%e It appears that rows converge to A194697.

%Y Cf. A139251, A160121, A160407, A161831, A194271, A194441, A194442, A194445, A194694, A194695, A194697.

%K nonn

%O 0,3

%A _Omar E. Pol_, Aug 29 2011