A139251 -id:A139251 - cp's OEIS frontend

A161207 First differences of A161206.

1, 2, 4, 6, 8, 10, 12, 14, 12, 12, 18, 24, 30, 30, 28, 30, 20, 12, 18, 26, 34, 42, 50, 56, 54, 44, 48, 64, 82, 80, 68, 66, 36, 12, 18, 26, 34, 42, 50, 58, 58, 54, 66, 90, 114, 126, 122, 120, 102, 60, 48, 70, 94, 118, 142, 160, 162, 136, 130, 160, 204, 198, 160, 142, 68, 12

Offset: 1

Omar E. Pol, Jun 08 2009

nonn

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

David Applegate, The movie version
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. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A139250, A139251, A160121, A160173, A161206, A161329, A161331.

More terms from R. J. Mathar, Jan 21 2010

A160407 First differences of toothpick numbers A160406.

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 4, 4, 2, 2, 4, 4, 4, 6, 10, 8, 2, 2, 4, 4, 4, 6, 10, 8, 4, 6, 10, 10, 12, 20, 26, 16, 2, 2, 4, 4, 4, 6, 10, 8, 4, 6, 10, 10, 12, 20, 26, 16, 4, 6, 10, 10, 12, 20, 26, 18, 12, 20, 28, 30, 42

Offset: 1

Omar E. Pol, May 23 2009

nonn

Number of toothpicks added at n-th stage in the toothpick structure of A160406.
From Omar E. Pol, Mar 15 2020: (Start)
The cellular automaton described in A160406 has word "ab", so the structure of this triangle 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 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.
For further information about the "word" of a cellular automaton see A296612. (End)

			From _Omar E. Pol_, Jul 18 2009, Mar 15 2020: (Start)
If written as a triangle:
1,1;
2,2;
2,2,4,4;
2,2,4,4,4,6,10,8;
2,2,4,4,4,6,10,8,4,6,10,10,12,20,26,16;
2,2,4,4,4,6,10,8,4,6,10,10,12,20,26,16,4,6,10,10,12,20,26,18,12,20,28,30,42;...
(End)

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. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A011782, A139250, A139251, A153000, A153006, A152980, A160406, A161830, A161831, A296612.

More terms from N. J. A. Sloane, Jul 17 2009

A161328 E-toothpick sequence (see Comments lines for definition).

Original entry on oeis.org

0, 1, 4, 9, 16, 29, 40, 57, 72, 93, 116, 141, 168, 201, 228, 253, 268, 293, 328, 369, 424, 477, 536, 597, 656, 721, 784, 841, 888, 925, 972, 1037, 1108, 1205, 1300, 1405, 1500, 1589, 1672, 1753, 1840, 1933, 2012, 2085, 2164, 2253, 2360, 2473, 2592, 2705, 2820

Offset: 0

Omar E. Pol, Jun 07 2009

nonn

An E-toothpick is formed by three toothpicks, as an trident. The E-toothpick has a midpoint and three exposed endpoints such that the distance between the endpoint of the central toothpick and the endpoints of the other toothpicks is equal to 1.
On the infinite triangular grid, we start at round 0 with no E-toothpicks.
At round 1 we place an E-toothpick anywhere in the plane.
At round 2 we add three more E-toothpicks.
At round 3 we add five more E-toothpicks.
And so on... (see illustrations).
The rule for adding new E-toothpicks is as follows. Each E has three ends, which initially are free. If the ends of two E's meet, those ends are no longer free. To go from round n to round n+1, we add an E-toothpick at each free end (extending that end in the direction it is pointing), subject to the condition that no end of any new E can touch any end of an existing E from round n or earlier. (Two new E's are allowed to touch.)
The sequence gives the number of E-toothpicks in the structure after n rounds. A161329 (the first differences) gives the number added at the n-th round.
Note that, on the infinite triangular grid, a E-toothpick can be represented as a polyedge with three components. In this case, at n-th round, the structure is a polyedge with 3*a(n) components. See the entry A139250 for more information about the growth of the toothpicks.
See also the snowflake sequence A161330.

David Applegate, The movie version
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. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
Dr. Goulu, 2012, Mayas, Kaya et cure-dents, Pourquoi Comment Combien blog, January 2012 (in French).
Omar E. Pol, A magic wand with star in the E-toothpick cellular automaton
Omar E. Pol, Illustration of initial terms of A160120, A161206, A161328, A161330
N. J. A. Sloane, A single E-toothpick
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Zozoped, Illustration of the structure, a(42) = 2012 [Broken link].
Zozoped, Illustration of the structure, a(42) = 2012, "Nous avons vu se lever son étoile", Le blog du Barabel [broken link].
Index entries for sequences related to toothpick sequences
Index entries for sequences related to cellular automata

Cf. A139250, A139251, A160120, A160172, A161206, A161329, A161330.

For n >= 3, a(n) = 4 + Sum_{k=3..n} 2*Sum_{x=1..3} A220498(k-x) + 2^((k mod 2) + 1) - 7. - Christopher Hohl, Feb 24 2019

a(8) corrected, more terms appended by R. J. Mathar, Jan 21 2010
Extensive edits by Omar E. Pol, May 14 2012
I have copied the rule for adding new E-toothpicks (described by N. J. A. Sloane) from A161330. - Omar E. Pol, Dec 07 2012

A220501 Number of toothpicks or D-toothpicks added at n-th stage in the structure of A220500.

Original entry on oeis.org

0, 1, 4, 8, 16, 22, 24, 24, 36, 40, 32, 44, 64, 94, 72, 56, 76, 72, 32, 48, 80, 112, 140, 140, 140, 168, 120, 132, 168, 238, 168, 120, 156, 136, 32, 48, 80, 112, 140, 144, 164, 224, 216, 256, 272, 368, 324, 316, 280, 296, 160, 150, 232, 320, 404, 432, 408, 462, 300, 308, 376, 526, 360, 248, 316, 264, 32

Offset: 0

Omar E. Pol, Dec 15 2012

Essentially the first differences of A220500.

			Written as an irregular triangle the sequence begins:
0;
1;
4;
8;
16,22;
24,24,36,40;
32,44,64,94,72,56,76,72;
32,48,80,112,140...
.
From _Omar E. Pol_, May 02 2018: (Start)
Also the nonzero terms can be written as an irregular triangle in which the row lengths are the terms of A011782 multiplied by 2 as shown below:
1,    4;
8,   16;
22,  24, 24, 36;
40,  32, 44, 64,  94,  72,  56,  76;
72,  32, 48, 80, 112, 140, 140, 140, 168, 120, 132, 168, 238, 168, 120, 156;
136, 32, 48, 80, 112, 140, 144, 164, 224, 216, 256, 272, 368, 324, 316, 280, ...
(End)

David Applegate, The movie version
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences

Cf. A047999, A139250, A139251, A194271, A194441, A194443, A194701, A220500, A220521, A220523, A220527.

Terms a(23) and beyond from David Applegate's movie version. - Omar E. Pol, May 02 2018

A160172 T-toothpick sequence (see Comments lines for definition).

Original entry on oeis.org

0, 1, 4, 9, 18, 27, 36, 49, 74, 95, 104, 117, 142, 167, 192, 229, 302, 359, 368, 381, 406, 431, 456, 493, 566, 627, 652, 689, 762, 835, 908, 1017, 1234, 1399, 1408, 1421, 1446, 1471, 1496, 1533, 1606, 1667, 1692, 1729, 1802, 1875, 1948, 2057, 2274, 2443, 2468

Offset: 0

Omar E. Pol, Jun 01 2009

A T-toothpick is formed from three toothpicks of equal length, in the shape of a T. There are three endpoints. We call the middle of the top toothpick the pivot point.
We start at round 0 with no T-toothpicks.
At round 1 we place a T-toothpick anywhere in the plane.
At round 2 we place three other T-toothpicks.
And so on...
The rule for adding a new T-toothpick is the following. A new T-toothpick is added at any exposed endpoint, with the pivot point touching the endpoint and so that the crossbar of the new toothpick is perpendicular to the exposed end.
The sequence gives the number of T-toothpicks after n rounds. A160173 (the first differences) gives the number added at the n-th round.
See the entry A139250 for more information about the toothpick process and the toothpick propagation.
On the infinite square grid a T-toothpick can be represented as a square polyedge with three components from a central point: two consecutive components on the same straight-line and a centered orthogonal component.
If the T-toothpick has three components then at the n-th round the structure is a polyedge with 3*a(n) components.
From Omar E. Pol, Mar 26 2011: (Start)
For formula and more information see the Applegate-Pol-Sloane paper, chapter 11, "T-shaped toothpicks". See also A160173.
Also, this sequence can be illustrated using another structure in which every T-toothpick is replaced by an isosceles right triangle. (End)
The structure is very distinct but the graph is similar to the graphs from the following sequences: A147562, A160164, A162795, A169707, A187220, A255366, A256260, at least for the known terms from Data section. - Omar E. Pol, Nov 24 2015
Shares with A255366 some terms with the same index, for example the element a(43) = 1729, the Hardy-Ramanujan number. - Omar E. Pol, Nov 25 2015

David Applegate, The movie version
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. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.],
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A139250, A139251, A147562, A160120, A160160, A160164, A160170, A160173, A160406, A160408, A160426, A160800, A162795, A169707, A187220, A255366, A256260.

wt[n_] := DigitCount[n, 2, 1];
A151920[n_] := Sum[3^wt[i], {i, 1, n + 1}]/3;
a[n_] := 2*A151920[n - 2] + 2*A151920[n - 3] + n;
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Apr 21 2024, after Charlie Neder *)

a(n) = 2*A151920(n) + 2*A151920(n-1) + n + 1. - Charlie Neder, Feb 07 2019

Edited and extended by N. J. A. Sloane, Jan 01 2010

A060632 a(n) = 2^wt(floor(n/2)) (i.e., 2^A000120(floor(n/2)), or A001316(floor(n/2))).

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 4, 4, 2, 2, 4, 4, 4, 4, 8, 8, 2, 2, 4, 4, 4, 4, 8, 8, 4, 4, 8, 8, 8, 8, 16, 16, 2, 2, 4, 4, 4, 4, 8, 8, 4, 4, 8, 8, 8, 8, 16, 16, 4, 4, 8, 8, 8, 8, 16, 16, 8, 8, 16, 16, 16, 16, 32, 32, 2, 2, 4, 4, 4, 4, 8, 8, 4, 4, 8, 8, 8, 8, 16, 16, 4, 4, 8, 8, 8, 8, 16, 16, 8, 8, 16, 16, 16, 16, 32

Offset: 0

Avi Peretz (njk(AT)netvision.net.il), Apr 15 2001

nonn

Number of conjugacy classes in the symmetric group S_n that have odd number of elements.
Also sequence A001316 doubled.
Number of even numbers whose binary expansion is a child of the binary expansion of n. - Nadia Heninger and N. J. A. Sloane, Jun 06 2008
First differences of A151566. Sequence gives number of toothpicks added at the n-th generation of the leftist toothpick sequence A151566. - N. J. A. Sloane, Oct 20 2010
The Fi1 and Fi1 triangle sums, see A180662 for their definitions, of Sierpiński's triangle A047999 equal this sequence. - Johannes W. Meijer, Jun 05 2011
Also number of odd entries in n-th row of triangle of Stirling numbers of the first kind. - Istvan Mezo, Jul 21 2017

			a(3) = 2 because in S_3 there are two conjugacy classes with odd number of elements, the trivial conjugacy class and the conjugacy class of transpositions consisting of 3 elements: (12),(13),(23).
From _Omar E. Pol_, Oct 12 2011 (Start):
Written as a triangle:
1,
1,
2,2,
2,2,4,4,
2,2,4,4,4,4,8,8,
2,2,4,4,4,4,8,8,4,4,8,8,8,8,16,16,
2,2,4,4,4,4,8,8,4,4,8,8,8,8,16,16,4,4,8,8,8,8,16,16,8,...
(End)

I. G. MacDonald: Symmetric functions and Hall polynomials Oxford: Clarendon Press, 1979. Page 21.

Indranil Ghosh, Table of n, a(n) for n = 0..65536 (terms 0..1000 from Harry J. Smith)
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. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
Christina Talar Bekaroğlu, Analyzing Dynamics of Larger than Life: Impacts of Rule Parameters on the Evolution of a Bug's Geometry, Master's thesis, Calif. State Univ. Northridge (2023). See p. 92.
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences

Cf. A000120, A001316, A139251, A151566, A160407.

a000120:=func< n | &+Intseq(n, 2) >; [ 2^a000120(Floor(n/2)): n in [0..100] ]; // Klaus Brockhaus, Oct 15 2010

A060632 := proc(n) local k; add(binomial(n,2*k) mod 2, k=0..floor(n/2)); end: seq(A060632(n),n=0..94); # edited by Johannes W. Meijer, May 28 2011
A060632 := n -> 2^add(i, i = convert(iquo(n,2), base, 2)); # Peter Luschny, Jun 30 2011
A060632 := n -> igcd(2^n, n! / iquo(n,2)!^2);  # Peter Luschny, Jun 30 2011

a[n_] := 2^DigitCount[Floor[n/2], 2, 1]; Table[a[n], {n, 0, 94}] (* Jean-François Alcover, Feb 25 2014 *)

for (n=0, 1000, write("b060632.txt", n, " ", sum(k=0, floor(n/2), binomial(n, 2*k) % 2)) ) \\ Harry J. Smith, Sep 14 2009

a(n)=2^hammingweight(n\2) \\ Charles R Greathouse IV, Feb 06 2017

def A060632(n):
    return 2**bin(n/2)[2:].count("1") # Indranil Ghosh, Feb 06 2017

a(n) = sum{k=0..floor(n/2), C(n, 2k) mod 2} - Paul Barry, Jan 03 2005, Edited by Harry J. Smith, Sep 15 2009
a(n) = gcd(A056040(n), 2^n). - Peter Luschny, Jun 30 2011
G.f.: (1 + x) * Product_{k>=0} (1 + 2*x^(2^(k+1))). - Ilya Gutkovskiy, Jul 19 2019

More terms from James Sellers, Apr 16 2001
Edited by N. J. A. Sloane, Jun 06 2008; Oct 11 2010
a(0) = 1 added by N. J. A. Sloane, Sep 14 2009
Formula corrected by Harry J. Smith, Sep 15 2009

A139252 Number of segments needed to draw the toothpick structure of A139250 as it is after n stages.

Original entry on oeis.org

0, 1, 3, 5, 7, 11, 15, 17, 19, 23, 27, 31, 39, 51, 59, 61, 63, 67, 71, 75, 83, 95, 103, 107, 115, 127, 139, 155, 183, 215, 231, 233, 235, 239, 243, 247, 255, 267, 275, 279, 287, 299, 311, 327, 355, 387, 403, 407, 415

Offset: 0

Omar E. Pol, May 17 2008

nonn

Contribution from Omar E. Pol, Sep 16 2012 (Start):
It appears that A147614(n)/a(n) converge to 4.
It appears that A139250(n)/a(n) converge to 3.
It appears that A160124(n)/a(n) converge to 2.
(End)

			For n = 3, after three stages the toothpick structure of A139250 contains seven toothpicks (A139250(3) = 7), however the toothpick structure can be essentially represented by five segments, so a(3) = 5. - _Omar E. Pol_, Sep 16 2012

Nathaniel Johnston, Table of n, a(n) for n = 0..256
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. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
Nathaniel Johnston, C script for computing table of terms
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A139250, A139251, A139253, A147614, A160124, A139254, A139255, A160128.

Terms after a(28) from Nathaniel Johnston, Mar 29 2011

A139253 Toothpick primes: primes in the toothpick sequence A139250.

Original entry on oeis.org

3, 7, 11, 23, 43, 47, 67, 79, 223, 251, 283, 347, 383, 571, 683, 719, 859, 1163, 1279, 1319, 1607, 1667, 1759, 1871, 2011, 2731, 2767, 2843, 2879, 3919, 4523, 4783, 4799, 4903, 5051, 6343, 6607, 7823, 8971, 9643, 11003, 11071, 11483, 11519, 12251, 12907

Offset: 1

Omar E. Pol, Apr 24 2008

nonn

All Wagstaff primes A000979 are members of this sequence. - Omar E. Pol, Mar 12 2012

			a(25) = 2011 because 2011 is the number of toothpicks after 60 stages in the toothpick structure of A139250 and it is also a prime number A000040.

R. J. Mathar, Table of n, a(n) for n = 1..181
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. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.], arXiv:1004.3036
Chris K. Caldwell and G. L. Honaker, Jr., Prime Curios! 2011
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences

Intersection of A000040 and A139250.

Cf. A000979, A139251, A139252, A194810.

toothpicks = With[{terms = 200}, CoefficientList[(x/((1-x)(1+2x)))(1 + 2x Product[1 + x^(2^k-1) + 2 x^(2^k), {k, 0, Log[2, terms] // Ceiling}]) + O[x]^terms, x]];
Select[toothpicks, PrimeQ] (* Jean-François Alcover, Aug 05 2018 *)

A000040 INTERSECT A139250.

More terms from Omar E. Pol, Dec 20 2008

Terms beyond 2731 from R. J. Mathar, Jan 21 2010

A160118 Number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton (see Comments for precise definition).

Original entry on oeis.org

0, 1, 9, 13, 41, 45, 73, 85, 169, 173, 201, 213, 297, 309, 393, 429, 681, 685, 713, 725, 809, 821, 905, 941, 1193, 1205, 1289, 1325, 1577, 1613, 1865, 1973, 2729, 2733, 2761, 2773, 2857, 2869, 2953, 2989, 3241, 3253, 3337, 3373, 3625, 3661, 3913, 4021, 4777, 4789

Offset: 0

Omar E. Pol, May 05 2009

nonn

On the infinite square grid, we start at stage 0 with all square cells in the OFF state.
Define a "peninsula cell" to a cell that is connected to the structure by exactly one of its vertices.
At stage 1 we turn ON a single cell in the central position.
For n>1, if n is even, at stage n we turn ON all the OFF neighboring cells from cells that were turned in ON at stage n-1.
For n>1, if n is odd, at stage n we turn ON all the peninsular OFF cells.
For the corresponding corner sequence, see A160796.
An animation will show the fractal-like behavior (cf. A139250).
For the first differences see A160415. - Omar E. Pol, Mar 21 2011
First differs from A188343 at a(13). - Omar E. Pol, Mar 28 2011

			If we label the generations of cells turned ON by consecutive numbers we get the cell pattern shown below:
9...............9
.888.888.888.888.
.878.878.878.878.
.8866688.8866688.
...656.....656...
.8866444.4446688.
.878.434.434.878.
.888.4422244.888.
.......212.......
.888.4422244.888.
.878.434.434.878.
.8866444.4446688.
...656.....656...
.8866688.8866688.
.878.878.878.878.
.888.888.888.888.
9...............9
In the first generation, only the central "1" is ON, a(1)=1. In the next generation, we turn ON eight "2" around the central cell, leading to a(2)=a(1)+8=9. In the third generation, four "3" are turned ON at the vertices of the square, a(3)=a(2)+4=13. And so on...

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. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.],
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS.
Index entries for sequences related to cellular automata.

Cf. A139250, A139251, A147562, A147610, A160117, A160119, A160379, A160415, A160796, A188343.

With[{d = 2}, wt[n_] := DigitCount[n, 2, 1]; a[n_] := If[OddQ[n], 3^d + (2^d)*Sum[(2^d - 1)^(wt[k] - 1), {k, 1, (n - 1)/2}] + (2^d)*(3^d - 2)*Sum[(2^d - 1)^(wt[k] - 1), {k, 1, (n - 3)/2}], 3^d + (2^d)*Sum[(2^d - 1)^(wt[k] - 1), {k, 1, n/2 - 1}] + (2^d)*(3^d - 2)*Sum[(2^d - 1)^(wt[k] - 1), {k, 1, n/2 - 1}]]; a[0] = 0; a[1] = 1; Array[a, 50, 0]] (* Amiram Eldar, Aug 01 2023 *)

From Nathaniel Johnston, Mar 24 2011: (Start)
a(2n-1) = 9 + 4*Sum_{k=2..n} A147610(k) + 28*Sum_{k=2..n-1} A147610(k), n >= 2.
a(2n) = 9 + 4*Sum_{k=2..n} A147610(k) + 28*Sum_{k=2..n} A147610(k), n >= 1.
(End)

Entry revised by Omar E. Pol and N. J. A. Sloane, Feb 16 2010, Feb 21 2010
a(8) - a(38) from Nathaniel Johnston, Nov 06 2010
a(13) corrected at the suggestion of Sean A. Irvine. Then I corrected 19 terms between a(14) and a(38). Finally I added a(39)-a(42). - Omar E. Pol, Mar 21 2011
Rule, for n even, edited by Omar E. Pol, Mar 22 2011
Incorrect comment (in "formula" section) removed by Omar E. Pol, Mar 23 2011, with agreement of author.
More terms from Amiram Eldar, Aug 01 2023

A194441 Number of toothpicks or D-toothpicks added at n-th stage to the structure of A194440.

Original entry on oeis.org

0, 1, 2, 4, 4, 4, 4, 8, 8, 4, 4, 8, 12, 16, 8, 16, 16, 4, 4, 8, 12, 16, 16, 24, 26, 24, 12, 20, 28, 40, 20, 32, 32, 4, 4, 8, 12, 16, 16, 24, 26, 24, 20, 32, 40, 64, 40, 48, 54, 40, 12, 20, 32, 48, 48, 64, 70, 76, 30, 44, 64, 88, 44, 64, 64, 4, 4, 8, 12

Offset: 0

Omar E. Pol, Aug 29 2011

nonn

Essentially the first differences of A194440.

			If written as a triangle:
0,
1,
2,
4,4,
4,4,8,8,
4,4,8,12,16,8,16,16,
4,4,8,12,16,16,24,26,24,12,20,28,40,20,32,32,
4,4,8,12,16,16,24,26,24,20,32,40,64,40,48,54,40,12,20,...
.
It appears that rows converge to A194696.

Cf. A139251, A160121, A160407, A161831, A194271, A194440, A194443, A194445, A194692, A194693, A194696.

Conjectures for n = 2^k+j, if -1<=j<=3:
a(2^k-1) = 2^k, if k >= 2.
a(2^k+0) = 2^k, if k >= 0.
a(2^k+1) = 4, if k >= 1.
a(2^k+2) = 4, if k >= 1.
a(2^k+3) = 8, if k >= 2.
End of conjectures.

More terms from Omar E. Pol, Dec 28 2012

cp's OEIS Frontend

A161207 First differences of A161206.

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A160407 First differences of toothpick numbers A160406.

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A161328 E-toothpick sequence (see Comments lines for definition).

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A220501 Number of toothpicks or D-toothpicks added at n-th stage in the structure of A220500.

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A160172 T-toothpick sequence (see Comments lines for definition).

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A060632 a(n) = 2^wt(floor(n/2)) (i.e., 2^A000120(floor(n/2)), or A001316(floor(n/2))).

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Magma

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PARI

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A139252 Number of segments needed to draw the toothpick structure of A139250 as it is after n stages.

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A139253 Toothpick primes: primes in the toothpick sequence A139250.

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