This site is supported by donations to The OEIS Foundation.

Annual Appeal: Please make a donation (tax deductible in USA) to keep the OEIS running. Over 4500 articles have referenced us, often saying "we would not have discovered this result without the OEIS".

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A139251 First differences of toothpicks numbers A139250. 211

%I

%S 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 36,40,60,88,80,32,4,8,12,12,16,28,32,20,16,28,36,40,60,88,80,36,16,

%U 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 First differences of toothpicks numbers A139250.

%C Number of toothpicks added to the toothpick structure at the n-th step (see A139250).

%C 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 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 Concerning this conjecture, see _David Applegate_'s comments on the conjectures in A153006. - _N. J. A. Sloane_, May 14 2009

%C 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 Equals (1, 2, 2, 2,...) convolved with A160762: (1, 0, 2, -2, 2, 2, 2, -6,...). - _Gary W. Adamson_, May 25 2009

%C Convolved with the Jacobsthal sequence A001045 = A160704: (1, 3, 9, 19, 41,...). - _Gary W. Adamson_, May 24 2009

%C It appears that the sums of two successive terms of A160552 give the positive terms of this sequence. - _Omar E. Pol_, Feb 19 2015

%H N. J. A. Sloane, <a href="/A139251/b139251.txt">Table of n, a(n) for n = 0..65535</a>

%H David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://neilsloane.com/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>, Congressus Numerantium, Vol. 206 (2010), 157-191

%H David Applegate, <a href="http://www.research.att.com/~david/oeis/toothpick.html">The movie version</a>

%H 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 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 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 Recurrence from _David Applegate_, Apr 29 2009: (Start)

%F Write n=2^(i+1)+j, where 0<=j<2^(i+1). Then, for n > 3:

%F for j=0, a(n) = 2*a(n-2^i) (= n = 2^(i+1))

%F for 1<=j<=2^i-1, a(n) = a(n-2^i)

%F for j=2^i, a(n) = a(n-2^i)+4 (= 2^(i+1)+4)

%F for 2^i+1<=j<=2^(i+1)-2, a(n) = 2*a(n-2^i)+a(n-2^i+1)

%F for j=2^(i+1)-1, a(n) = 2*a(n-2^i)+a(n-2^i+1)-4

%F and a(n) = 2^(n-1) for n=1,2,3. (End)

%F G.f.: (x/(1+2*x)) * (1 + 2*x*Product(1+x^(2^k-1)+2*x^(2^k),k=0..oo)). - _N. J. A. Sloane_, May 20 2009, Jun 05 2009

%F 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 It appears that a(n) = A187221(n+1)/2. - _Omar E. Pol_, Mar 08 2011

%F It appears that a(n) = A160552(n-1) + A160552(n), n >= 1. - _Omar E. Pol_, Feb 18 2015

%e Triangle begins:

%e . 0;

%e . 1;

%e . 2,4;

%e . 4,4,8,12;

%e . 8,4,8,12,12,16,28,32;

%e .16,4,8,12,12,16,28,32,20,16,28,36,40,60,88,80;

%e .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 ...

%e The row sums give A006516.

%p 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 # A139250 is T, A139251 is a.

%p a:=[0,1,2,4]; T:=[0,1,3,7]; M:=10;

%p for k from 1 to M do

%p a:=[op(a),2^(k+1)];

%p T:=[op(T),T[nops(T)]+a[nops(a)]];

%p for j from 1 to 2^(k+1)-1 do

%p a:=[op(a), 2*a[j+1]+a[j+2]];

%p T:=[op(T),T[nops(T)]+a[nops(a)]];

%p od: od: a; T;

%p # _N. J. A. Sloane_, Dec 25 2009

%t 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 Equals 2*A152968 and 4*A152978 (if we ignore the first couple of terms).

%Y See A147646 for the limiting behavior of the rows. See also A006516.

%Y Row lengths in A011782.

%Y Cf. A139250, A139252, A139253, A152980, A153000, A153001, A000396, A000668, A061652, A153006.

%Y Cf. A006516, A153007, A159790, A001045, A160704, A160762, A160121, A147582.

%K nonn,tabf,look

%O 0,3

%A _Omar E. Pol_, Apr 24 2008, Dec 16 2008, Apr 20 2009

%E The layout of the triangle was adjusted by _David Applegate_, Apr 29 2009, to reveal that the columns become constant.

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .