

A139251


First differences of toothpicks numbers A139250.


232



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, 36, 40, 60, 88, 80, 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, 60, 92, 112, 140, 208, 256, 192, 64, 4, 8, 12, 12, 16, 28, 32, 20, 16, 28
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OFFSET

0,3


COMMENTS

Number of toothpicks added to the toothpick structure at the nth step (see A139250).
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.)
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 kth 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
Concerning this conjecture, see David Applegate's comments on the conjectures in A153006.  N. J. A. Sloane, May 14 2009
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
Equals (1, 2, 2, 2, ...) convolved with A160762: (1, 0, 2, 2, 2, 2, 2, 6, ...).  Gary W. Adamson, May 25 2009
Convolved with the Jacobsthal sequence A001045 = A160704: (1, 3, 9, 19, 41, ...).  Gary W. Adamson, May 24 2009
It appears that the sums of two successive terms of A160552 give the positive terms of this sequence.  Omar E. Pol, Feb 19 2015
From Omar E. Pol, Feb 28 2019: (Start)
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.
The structure of this triangle (with word "ab" and k = 2) for the nonzero terms is as follows:
a,b;
a,b;
a,b,a,b;
a,b,a,b,a,b,a,b;
a,b,a,b,a,b,a,b,a,b,a,b,a,b,a,b;
...
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, ...
This arrangement has the property that the oddindexed columns (a) contain numbers of the toothpicks that are parallel to initial toothpick, and the evenindexed columns (b) contain numbers of the toothpicks that are orthogonal to the initial toothpick (see the third triangle in the Example section).
An associated sound to the animation could be (tick, tock), (tick, tock), ..., the same as the ticking clock sound.
For further information about the "word" of a cellular automaton see A296612. (End)


LINKS

N. J. A. Sloane, Table of n, a(n) for n = 0..65535
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n1)1) for n >= 2.], Congressus Numerantium, Vol. 206 (2010), 157191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n1)1) for n >= 2.]
David Applegate, The movie version
Omar E. Pol, Illustration of initial terms of A139251, A160121, A147582 (Overlapping figures) [From Omar E. Pol, Nov 02 2009]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS


FORMULA

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
Recurrence from David Applegate, Apr 29 2009: (Start)
Write n=2^(i+1)+j, where 0 <= j < 2^(i+1). Then, for n > 3:
for j=0, a(n) = 2*a(n2^i) (= n = 2^(i+1))
for 1 <= j <= 2^i  1, a(n) = a(n2^i)
for j=2^i, a(n) = a(n2^i)+4 (= 2^(i+1)+4)
for 2^i+1 <= j <= 2^(i+1)2, a(n) = 2*a(n2^i) + a(n2^i+1)
for j=2^(i+1)1, a(n) = 2*a(n2^i) + a(n2^i+1)4
and a(n) = 2^(n1) for n=1,2,3. (End)
G.f.: (x/(1+2*x)) * (1 + 2*x*Product_{k>=0} (1 + x^(2^k1) + 2*x^(2^k))).  N. J. A. Sloane, May 20 2009, Jun 05 2009
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
It appears that a(n) = A187221(n+1)/2.  Omar E. Pol, Mar 08 2011
It appears that a(n) = A160552(n1) + A160552(n), n >= 1.  Omar E. Pol, Feb 18 2015


EXAMPLE

From Omar E. Pol, Dec 16 2008: (Start)
Triangle begins:
1;
2;
4,4;
4,8,12,8;
4,8,12,12,16,28,32,16;
4,8,12,12,16,28,32,20,16,28,36,40,60,88,20,32;
(End)
From David Applegate, Apr 29 2009: (Start)
The layout of the triangle was adjusted to reveal that the columns become constant as shown below:
. 0;
. 1;
. 2,4;
. 4,4,8,12;
. 8,4,8,12,12,16,28,32;
.16,4,8,12,12,16,28,32,20,16,28,36,40,60,88,80;
.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,...
...
The row sums give A006516.
(End)
From Omar E. Pol, Feb 28 2018: (Start)
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:
1,2;
4,4;
4,8,12,8;
4,8,12,12,16,28,32,16;
4,8,12,12,16,28,32,20,16,28,36,40,60,88,20,32;
...
(End)


MAPLE

G := (x/(1+2*x)) * (1 + 2*x*mul(1+x^(2^k1)+2*x^(2^k), k=0..20)); # N. J. A. Sloane, May 20 2009, Jun 05 2009
# A139250 is T, A139251 is a.
a:=[0, 1, 2, 4]; T:=[0, 1, 3, 7]; M:=10;
for k from 1 to M do
a:=[op(a), 2^(k+1)];
T:=[op(T), T[nops(T)]+a[nops(a)]];
for j from 1 to 2^(k+1)1 do
a:=[op(a), 2*a[j+1]+a[j+2]];
T:=[op(T), T[nops(T)]+a[nops(a)]];
od: od: a; T;
# N. J. A. Sloane, Dec 25 2009


MATHEMATICA

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 *)


CROSSREFS

Equals 2*A152968 and 4*A152978 (if we ignore the first couple of terms).
See A147646 for the limiting behavior of the rows. See also A006516.
Row lengths in A011782.
Cf. A139250, A139252, A139253, A152980, A153000, A153001, A000396, A000668, A061652, A153006.
Cf. A006516, A153007, A159790, A001045, A160704, A160762, A147582, A296612.
Cf. A160121 (word "a"), A296511 (word "abc"), A299477 (word "abcb"), A299479 (word "abcbc").
Sequence in context: A194445 A220525 A160809 * A182635 A188346 A309690
Adjacent sequences: A139248 A139249 A139250 * A139252 A139253 A139254


KEYWORD

nonn,tabf,look


AUTHOR

Omar E. Pol, Apr 24 2008


EXTENSIONS

Partially edited by Omar E. Pol, Feb 28 2019


STATUS

approved



