A160407 -id:A160407 - cp's OEIS frontend

A160418 a(n) = A160407(n+2)/2.

1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 3, 5, 4, 1, 1, 2, 2, 2, 3, 5, 4, 2, 3, 5, 5, 6, 10, 13, 8, 1, 1, 2, 2, 2, 3, 5, 4, 2, 3, 5, 5, 6, 10, 13, 8, 2, 3, 5, 5, 6, 10, 13, 9, 6, 10, 14, 15, 21, 32, 33, 16, 1, 1, 2, 2, 2, 3, 5, 4, 2, 3, 5, 5, 6, 10, 13, 8, 2, 3, 5, 5

Offset: 1

Omar E. Pol, May 23 2009

Row lengths are the terms of A000079 multiplied by 2. Right border gives A000079. - Omar E. Pol, Mar 19 2020

			From _Omar E. Pol_, Mar 19 2020: (Start)
Triangle begins:
  1,1;
  1,1,2,2;
  1,1,2,2,2,3,5,4;
  1,1,2,2,2,3,5,4,2,3,5,5,6,10,13,8;
  1,1,2,2,2,3,5,4,2,3,5,5,6,10,13,8,2,3,5,5,6,10,13,9,6,10,14,15,21,32,33,16;
  ... (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. A139250, A139251, A160160, A160161, A160406, A160407, A160408, A160409.

More terms from Jinyuan Wang, Mar 14 2020

A160406 Toothpick sequence starting at the vertex of an infinite 90-degree wedge.

Original entry on oeis.org

0, 1, 2, 4, 6, 8, 10, 14, 18, 20, 22, 26, 30, 34, 40, 50, 58, 60, 62, 66, 70, 74, 80, 90, 98, 102, 108, 118, 128, 140, 160, 186, 202, 204, 206, 210, 214, 218, 224, 234, 242, 246, 252, 262, 272, 284, 304, 330, 346, 350, 356, 366, 376, 388, 408, 434, 452, 464, 484, 512, 542, 584

Offset: 0

Omar E. Pol, May 23 2009

nonn

Consider the wedge of the plane defined by points (x,y) with y >= |x|, with the initial toothpick extending from (0,0) to (0,2); then extend by the same rule as for A139250, always staying inside the wedge.
Number of toothpick in the structure after n rounds.
The toothpick sequence A139250 is the main entry for this sequence. See also A153000. First differences: A160407.

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.]
Omar E. Pol, Illustration of initial terms
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A139250, A139251, A153000, A153006, A152980, A160407, A160408, A160409.

Cf. A170886-A170895.

G := (x + 2*x^2 + 4*x^2*(1+x)*(mul(1+x^(2^k-1)+2*x^(2^k),k=1..20)-1)/(1+2*x))/(1-x); P:=(G + 2 + x*(5-x)/(1-x)^2)*x/(2*(1+x)); series(P,x,200); seriestolist(%); # N. J. A. Sloane, May 25 2009

terms = 62;
G = (x + 2x^2 + 4x^2 (1+x)(Product[1+x^(2^k-1) + 2x^(2^k), {k, 1, Ceiling[ Log[2, terms]]}]-1)/(1+2x))/(1-x);
P = (G + 2 + x(5-x)/(1-x)^2) x/(2(1+x));
CoefficientList[P + O[x]^terms, x] (* Jean-François Alcover, Nov 03 2018, from Maple *)

A139250(n) = 2a(n) + 2a(n+1) - 4n - 1 for n > 0. - N. J. A. Sloane, May 25 2009

Let G = (x + 2*x^2 + 4*x^2*(1+x)*((Product_{k>=1} (1 + x^(2^k-1) + 2*x^(2^k))) - 1)/(1+2*x))/(1-x) (= g.f. for A139250); then the g.f. for the present sequence is (G + 2 + x*(5-x)/(1-x)^2)*x/(2*(1+x)). - N. J. A. Sloane, May 25 2009

More terms from N. J. A. Sloane, May 25 2009

Definition revised by N. J. A. Sloane, Jan 02 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

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

A194443 Number of toothpicks or D-toothpicks added at n-th stage to the structure of A194442.

Original entry on oeis.org

0, 1, 2, 4, 4, 4, 4, 7, 8, 4, 4, 8, 12, 8, 8, 13, 16, 4, 4, 8, 12, 16, 16, 20, 24, 12, 8, 16, 28, 16, 16, 25, 32, 4, 4, 8, 12, 16, 16, 22, 32, 26, 20, 24, 40, 32, 40, 33, 48, 20, 8, 16, 28, 40, 44, 50, 60, 28, 16, 32, 60, 32, 32, 49, 64, 4, 4, 8

Offset: 0

Omar E. Pol, Aug 29 2011

nonn

Essentially the first differences of A194442. It appears that the structure of the "narrow" triangle is much more regular about n=2^k, see formula section.

			If written as a triangle:
0,
1,
2,
4,4,
4,4,7,8,
4,4,8,12,8,8,13,16,
4,4,8,12,16,16,20,24,12,8,16,28,16,16,25,32,
4,4,8,12,16,16,22,32,26,20,24,40,32,40,33,48,20,8,16,28...
.
It appears that rows converge to A194697.

Cf. A139251, A160121, A160407, A161831, A194271, A194441, A194442, A194445, A194694, A194695, A194697.

Conjectures for n = 2^k+j, if -6<=j<=6:
a(2^k-6) = 2^(k-2), if k >= 3.
a(2^k-5) = 2^(k-1), if k >= 3.
a(2^k-4) = 2^k-4, if k >= 2.
a(2^k-3) = 2^(k-1), if k >= 3.
a(2^k-2) = 2^(k-1), if k >= 2.
a(2^k-1) = 3*2^(k-2)+1, 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 >= 3.
a(2^k+4) = 12, if k >= 3.
a(2^k+5) = 16, if k >= 4.
a(2^k+6) = 16, if k >= 4.
End of conjectures.

A079314 Number of first-quadrant cells (including the two boundaries) born at stage n of the Holladay-Ulam cellular automaton.

Original entry on oeis.org

1, 2, 2, 4, 2, 4, 4, 10, 2, 4, 4, 10, 4, 10, 10, 28, 2, 4, 4, 10, 4, 10, 10, 28, 4, 10, 10, 28, 10, 28, 28, 82, 2, 4, 4, 10, 4, 10, 10, 28, 4, 10, 10, 28, 10, 28, 28, 82, 4, 10, 10, 28, 10, 28, 28, 82, 10, 28, 28, 82, 28, 82, 82, 244, 2, 4, 4, 10, 4, 10, 10, 28, 4, 10, 10, 28, 10, 28, 28, 82, 4

Offset: 0

N. J. A. Sloane, Feb 12 2003

See the main entry for this CA, A147562, for further information.
When I first read the Singmaster MS in 2003 I misunderstood the definition of the CA. In fact once cells are ON they stay ON. The other version, when cells can change state from ON to OFF, is described in A079317. - N. J. A. Sloane, Aug 05 2009
The pattern has 4-fold symmetry; sequence just counts cells in one quadrant.

			From _Omar E. Pol_, Jul 18 2009: (Start)
If written as a triangle:
  1;
  2;
  2,4;
  2,4,4,10;
  2,4,4,10,4,10,10,28;
  2,4,4,10,4,10,10,28,4,10,10,28,10,28,28,82;
  2,4,4,10,4,10,10,28,4,10,10,28,10,28,28,82,4,10,10,28,10,28,28,82,10,28;...
Rows converge to A151712.
(End)

D. Singmaster, On the cellular automaton of Ulam and Warburton, M500 Magazine of the Open University, #195 (December 2003), pp. 2-7.

Paolo Xausa, Table of n, a(n) for n = 0..10000
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.]
Omar E. Pol, Illustration of initial terms (Overlapping squares) [From _Omar E. Pol_, Nov 20 2009]
D. Singmaster, On the cellular automaton of Ulam and Warburton, 2003 [Cached copy, included with permission]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168, 2015

Cf. A147582, A079315, A079316, A079317, A079318, A079319, A151713, A139250.

Cf. A151712, A151922, A160407.

A079314list[nmax_]:=Join[{1},3^(DigitCount[Range[nmax],2,1]-1)+1];A079314list[100] (* Paolo Xausa, Jun 29 2023 *)

For n > 0, a(n) = 3^(A000120(n)-1) + 1.
For n > 0, a(n) = A147582(n)/4 + 1.
Partial sums give A151922. [Omar E. Pol, Nov 20 2009]

Edited by N. J. A. Sloane, Aug 05 2009

A161831 First differences of A161830.

Original entry on oeis.org

1, 2, 2, 4, 2, 4, 4, 8, 4, 4, 4, 8, 6, 8, 10, 18, 10, 4

Offset: 1

Omar E. Pol, Jun 20 2009

Number of Y-toothpicks added to the sieve at the n-th round.

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, A160407, A161830, A161832, A161833.

A161830 Y-toothpick triangle (see Comments lines for definition).

Original entry on oeis.org

0, 1, 3, 5, 9, 11, 15, 19, 27, 31, 35, 39, 47, 53, 61, 71, 89, 99, 103

Offset: 0

Omar E. Pol, Jun 20 2009

Y-toothpick sequence starting at the corner of an infinite hexagon in which its vertex touch an endpoint of the initial Y-toothpick and the two other endpoints are equidistant from the nearest sides of the hexagon.
The sequence gives the number of Y-toothpicks in the structure after n rounds. A161831 (the first differences) gives the number added at the n-th round.
See the Y-toothpick sequence A160120 for more information about the recursive, fractal-like structure.

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, A160120, A160121, A160406, A160407, A161831, A161832, A161833.

A160408 Toothpick pyramid (see Comments lines for definition).

Original entry on oeis.org

0, 1, 2, 4, 8, 12, 16, 20, 24, 32, 48, 64, 72, 76, 80, 88

Offset: 0

Omar E. Pol, May 23 2009, Jun 06 2009

The toothpick pyramid is a three-dimensional version of the toothpick triangle A160406.
The sequence gives the number of toothpicks after n rounds. A160409 (the first differences) gives the number added at the n-th round.
See also the entry A139250 for more information about the toothpick sequences.

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, A160160, A160406, A160407, A160409.

A160409 First differences of toothpick numbers A160408.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, May 23 2009

Number of toothpick added to the toothpick pyramid at the round n.

See also the toothpick sequences A139250, A160160 and the toothpick triangle A160406.

			Contribution from _Omar E. Pol_, Jun 06 2009: (Start)
Array begins:
========
x, y, z
========
1, 1, 2;
4, 4, 4;
4, 4, 8;
16, 16, 8;
4, 4, 8;
(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. A139250, A139251, A160160, A160161, A160406, A160407, A160408.

More terms from Omar E. Pol, Jun 06 2009

cp's OEIS Frontend

A160418 a(n) = A160407(n+2)/2.

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A160406 Toothpick sequence starting at the vertex of an infinite 90-degree wedge.

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

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A194443 Number of toothpicks or D-toothpicks added at n-th stage to the structure of A194442.

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Examples

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Formula

A079314 Number of first-quadrant cells (including the two boundaries) born at stage n of the Holladay-Ulam cellular automaton.

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Mathematica

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A161831 First differences of A161830.

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A161830 Y-toothpick triangle (see Comments lines for definition).