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A139251 First differences of toothpicks numbers A139250. 213
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Number of toothpicks added to the toothpick structure at the n-th 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 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

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

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), 157-191

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(n-2^i) (= n = 2^(i+1))

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

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

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

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

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

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

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(n-1) + A160552(n), n >= 1. - Omar E. Pol, Feb 18 2015

EXAMPLE

Triangle begins:

. 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.

MAPLE

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

# 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, A160121, A147582.

Sequence in context: A194445 A220525 A160809 * A182635 A188346 A173531

Adjacent sequences:  A139248 A139249 A139250 * A139252 A139253 A139254

KEYWORD

nonn,tabf,look

AUTHOR

Omar E. Pol, Apr 24 2008, Dec 16 2008, Apr 20 2009

EXTENSIONS

The layout of the triangle was adjusted by David Applegate, Apr 29 2009, to reveal that the columns become constant.

STATUS

approved

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Last modified March 27 06:50 EDT 2017. Contains 284144 sequences.