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A153006 Toothpick sequence starting at the outside corner of an infinite square from which protrudes a half toothpick.

%I A153006 #46 Jun 14 2021 15:15:32
%S A153006 0,1,3,6,9,13,20,28,33,37,44,53,63,78,100,120,129,133,140,149,159,174,
%T A153006 196,217,231,246,269,297,332,384,448,496,513,517,524,533,543,558,580,
%U A153006 601,615,630,653,681,716,768,832,881,903,918,941
%N A153006 Toothpick sequence starting at the outside corner of an infinite square from which protrudes a half toothpick.
%C A153006 a(n) is the total number of integer toothpicks after n steps.
%C A153006 It appears that this sequence is related to triangular numbers, Mersenne primes and even perfect numbers. Conjecture: a(A000668(n))=A000217(A000668(n)). Conjecture: a(A000668(n))=A000396(n), assuming there are no odd perfect numbers.
%C A153006 The main entry for this sequence is A139250. See also A152980 (the first differences) and A147646.
%C A153006 The Mersenne prime conjectures are true, but aren't really about Mersenne primes. a(2^i-1) = 2^i (2^i-1)/2 for all i (whether or not i or 2^i-1 is prime). This follows from the formulas for A139250(2^i-1) and A139250(2^i). - _David Applegate_, May 11 2009
%C A153006 Then we can write a(A000225(k)) = A006516(k), for k > 0. - _Omar E. Pol_, May 23 2009
%C A153006 Equals A151550 convolved with [1, 2, 2, 2, ...]. (This is equivalent to the observation that the g.f. is x((1+x)/(1-x)) * Product_{n >= 1} (1 + x^(2^n-1) + 2*x^(2^n)).) Equivalently, equals A151555 convolved with A151575. - _Gary W. Adamson_, May 25 2009
%C A153006 It appears that a(n) is also 1/4 of the total path length of a toothpick structure as A139250 after n-th stage which is constructed following a special rule: toothpicks of the new generation have length 4 when are placed on the square grid (every toothpick has four components of length 1), but after every stage, one (or two) of the four components of every toothpick of the new generation is removed, if such component contains a endpoint of the toothpick and if such endpoint is touching the midpoint or the endpoint of another toothpick. The truncated endpoints of the toothpicks remain exposed forever. Note that there are three sizes of toothpicks in the structure: toothpicks of length 4, 3 and 2. a(n) is also 1/4 of the number of grid points that are covered after n-th stage, except the central point of the structure. A159795 gives the total path length and also the total number of components in the structure after n-th stage. - _Omar E. Pol_, Oct 24 2011
%H A153006 N. J. A. Sloane, <a href="/A153006/b153006.txt">Table of n, a(n) for n = 0..16387</a>
%H A153006 David Applegate, <a href="/A139250/a139250.anim.html">The movie version</a>
%H A153006 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 A153006 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/poltp028.jpg"> Illustration of initial terms</a>
%H A153006 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/poltp496.jpg"> Illustration of A153006(31) = 496</a>
%H A153006 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 A153006 <a href="/index/To#toothpick">Index entries for sequences related to toothpick sequences</a>
%H A153006 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%F A153006 G.f.: x*((1 + x)/(1 - x)) * Product_{n >= 1} (1 + x^(2^n-1) + 2*x^(2^n)). - _N. J. A. Sloane_, May 20 2009
%p A153006 G:=x*((1 + x)/(1 - x)) * mul( (1 + x^(2^n-1) + 2*x^(2^n)), n=1..20); # _N. J. A. Sloane_, May 20 2009
%Y A153006 Cf. A000079, A000217, A000396, A000668, A139250, A139251, A139560, A152978, A152979, A152980, A153000, A153001, A153002.
%Y A153006 Cf. A000225, A006516.
%Y A153006 Cf. A153007, A078008, A151555.
%K A153006 nonn
%O A153006 0,3
%A A153006 _Omar E. Pol_, Dec 17 2008, Dec 19 2008, Apr 28 2009
%E A153006 Edited by _N. J. A. Sloane_, Dec 19 2008