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A139251 First differences of toothpicks numbers A139250.

%I A139251 #90 Feb 24 2021 02:48:18
%S A139251 0,1,2,4,4,4,8,12,8,4,8,12,12,16,28,32,16,4,8,12,12,16,28,32,20,16,28,
%T A139251 36,40,60,88,80,32,4,8,12,12,16,28,32,20,16,28,36,40,60,88,80,36,16,
%U A139251 28,36,40,60,88,84,56,60,92,112,140,208,256,192,64,4,8,12,12,16,28,32,20,16,28
%N A139251 First differences of toothpicks numbers A139250.
%C A139251 Number of toothpicks added to the toothpick structure at the n-th step (see A139250).
%C A139251 It appears that if n is equal to 1 plus a power of 2 with positive exponent then a(n) = 4. (For proof see the second Applegate link.)
%C A139251 It appears that there is a relation between this sequence, even superperfect numbers, Mersenne primes and even perfect numbers. Conjecture: The sum of the toothpicks added to the toothpick structure between the stage A061652(k) and the stage A000668(k) is equal to the k-th even perfect number, for k >= 1. For example: A000396(1) = 2+4 = 6. A000396(2) = 4+4+8+12 = 28. A000396(3) = 16+4+8+12+12+16+28+32+20+16+28+36+40+60+88+80 = 496. - _Omar E. Pol_, May 04 2009
%C A139251 Concerning this conjecture, see _David Applegate_'s comments on the conjectures in A153006. - _N. J. A. Sloane_, May 14 2009
%C A139251 In the triangle (See example lines), the sum of row k is equal to A006516(k), for k >= 1. - _Omar E. Pol_, May 15 2009
%C A139251 Equals (1, 2, 2, 2, ...) convolved with A160762: (1, 0, 2, -2, 2, 2, 2, -6, ...). - _Gary W. Adamson_, May 25 2009
%C A139251 Convolved with the Jacobsthal sequence A001045 = A160704: (1, 3, 9, 19, 41, ...). - _Gary W. Adamson_, May 24 2009
%C A139251 It appears that the sums of two successive terms of A160552 give the positive terms of this sequence. - _Omar E. Pol_, Feb 19 2015
%C A139251 From _Omar E. Pol_, Feb 28 2019: (Start)
%C A139251 The study of the toothpick automaton on triangular grid (A296510), and other C.A. of the same family, reveals that some cellular automata that have recurrent periods can be represented in general by irregular triangles (of first differences) whose row lengths are the terms of A011782 multiplied by k, where k >= 1, is the length of an internal cycle. This internal cycle is called "word" of a cellular automaton. For example: A160121 has word "a", so k = 1. This sequence has word "ab", so k = 2. A296511 has word "abc", so k = 3. A299477 has word "abcb" so k = 4. A299479 has word "abcbc", so k = 5.
%C A139251 The structure of this triangle (with word "ab" and k = 2) for the nonzero terms is as follows:
%C A139251 a,b;
%C A139251 a,b;
%C A139251 a,b,a,b;
%C A139251 a,b,a,b,a,b,a,b;
%C A139251 a,b,a,b,a,b,a,b,a,b,a,b,a,b,a,b;
%C A139251 ...
%C A139251 The row lengths are the terms of A011782 multiplied by 2, equaling the column 2 of the square array A296612: 2, 2, 4, 8, 16, ...
%C A139251 This arrangement has the property that the odd-indexed columns (a) contain numbers of the toothpicks that are parallel to initial toothpick, and the even-indexed columns (b) contain numbers of the toothpicks that are orthogonal to the initial toothpick (see the third triangle in the Example section).
%C A139251 An associated sound to the animation could be (tick, tock), (tick, tock), ..., the same as the ticking clock sound.
%C A139251 For further information about the "word" of a cellular automaton see A296612. (End)
%H A139251 N. J. A. Sloane, <a href="/A139251/b139251.txt">Table of n, a(n) for n = 0..65535</a>
%H A139251 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.], 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 A139251 David Applegate, <a href="/A139250/a139250.anim.html">The movie version</a>
%H A139251 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polca002.jpg">Illustration of initial terms of A139251, A160121, A147582 (Overlapping figures)</a> [From _Omar E. Pol_, Nov 02 2009]
%H A139251 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>
%F A139251 Recurrence from _N. J. A. Sloane_, Jul 20 2009: a(0) = 0; a(2^i)=2^i for all i; otherwise write n=2^i+j, 0 < j < 2^i, then a(n) = 2a(j)+a(j+1). Proof: This is a simplification of the following recurrence of _David Applegate_. QED
%F A139251 Recurrence from _David Applegate_, Apr 29 2009: (Start)
%F A139251 Write n=2^(i+1)+j, where 0 <= j < 2^(i+1). Then, for n > 3:
%F A139251 for j=0, a(n) = 2*a(n-2^i) (= n = 2^(i+1))
%F A139251 for 1 <= j <= 2^i - 1, a(n) = a(n-2^i)
%F A139251 for j=2^i, a(n) = a(n-2^i)+4 (= 2^(i+1)+4)
%F A139251 for 2^i+1 <= j <= 2^(i+1)-2, a(n) = 2*a(n-2^i) + a(n-2^i+1)
%F A139251 for j=2^(i+1)-1, a(n) = 2*a(n-2^i) + a(n-2^i+1)-4
%F A139251 and a(n) = 2^(n-1) for n=1,2,3. (End)
%F A139251 G.f.: (x/(1+2*x)) * (1 + 2*x*Product_{k>=0} (1 + x^(2^k-1) + 2*x^(2^k))). - _N. J. A. Sloane_, May 20 2009, Jun 05 2009
%F A139251 With offset 0 (which would be more natural, but offset 1 is now entrenched): a(0) = 1, a(1) = 2; for i >= 1, a(2^i) = 4; otherwise write n = 2^i +j, 0 < j < 2^i, then a(n) = 2 * Sum_{ k >= 0 } 2^(wt(j+k)-k)*binomial(wt(j+k),k). - _N. J. A. Sloane_, Jun 03 2009
%F A139251 It appears that a(n) = A187221(n+1)/2. - _Omar E. Pol_, Mar 08 2011
%F A139251 It appears that a(n) = A160552(n-1) + A160552(n), n >= 1. - _Omar E. Pol_, Feb 18 2015
%e A139251 From _Omar E. Pol_, Dec 16 2008: (Start)
%e A139251 Triangle begins:
%e A139251 1;
%e A139251 2;
%e A139251 4,4;
%e A139251 4,8,12,8;
%e A139251 4,8,12,12,16,28,32,16;
%e A139251 4,8,12,12,16,28,32,20,16,28,36,40,60,88,20,32;
%e A139251 (End)
%e A139251 From _David Applegate_, Apr 29 2009: (Start)
%e A139251 The layout of the triangle was adjusted to reveal that the columns become constant as shown below:
%e A139251 . 0;
%e A139251 . 1;
%e A139251 . 2,4;
%e A139251 . 4,4,8,12;
%e A139251 . 8,4,8,12,12,16,28,32;
%e A139251 .16,4,8,12,12,16,28,32,20,16,28,36,40,60,88,80;
%e A139251 .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 A139251 ...
%e A139251 The row sums give A006516.
%e A139251 (End)
%e A139251 From _Omar E. Pol_, Feb 28 2018: (Start)
%e A139251 Also the nonzero terms can write as an irregular triangle in which the row lengths are the terms of A011782 multiplied by 2 as shown below:
%e A139251 1,2;
%e A139251 4,4;
%e A139251 4,8,12,8;
%e A139251 4,8,12,12,16,28,32,16;
%e A139251 4,8,12,12,16,28,32,20,16,28,36,40,60,88,20,32;
%e A139251 ...
%e A139251 (End)
%p A139251 G := (x/(1+2*x)) * (1 + 2*x*mul(1+x^(2^k-1)+2*x^(2^k),k=0..20)); # _N. J. A. Sloane_, May 20 2009, Jun 05 2009
%p A139251 # A139250 is T, A139251 is a.
%p A139251 a:=[0,1,2,4]; T:=[0,1,3,7]; M:=10;
%p A139251 for k from 1 to M do
%p A139251 a:=[op(a),2^(k+1)];
%p A139251 T:=[op(T),T[nops(T)]+a[nops(a)]];
%p A139251 for j from 1 to 2^(k+1)-1 do
%p A139251 a:=[op(a), 2*a[j+1]+a[j+2]];
%p A139251 T:=[op(T),T[nops(T)]+a[nops(a)]];
%p A139251 od: od: a; T;
%p A139251 # _N. J. A. Sloane_, Dec 25 2009
%t A139251 CoefficientList[Series[((x - x^2)/((1 - x) (1 + 2 x))) (1 + 2 x Product[1 + x^(2^k - 1) + 2 x^(2^k), {k, 0, 20}]), {x, 0, 60}], x] (* _Vincenzo Librandi_, Aug 22 2014 *)
%Y A139251 Equals 2*A152968 and 4*A152978 (if we ignore the first couple of terms).
%Y A139251 See A147646 for the limiting behavior of the rows. See also A006516.
%Y A139251 Row lengths in A011782.
%Y A139251 Cf. A139250, A139252, A139253, A152980, A153000, A153001, A000396, A000668, A061652, A153006.
%Y A139251 Cf. A006516, A153007, A159790, A001045, A160704, A160762, A147582, A296612.
%Y A139251 Cf. A160121 (word "a"), A296511 (word "abc"), A299477 (word "abcb"), A299479 (word "abcbc").
%K A139251 nonn,tabf,look
%O A139251 0,3
%A A139251 _Omar E. Pol_, Apr 24 2008
%E A139251 Partially edited by _Omar E. Pol_, Feb 28 2019