A160409 -id:A160409 - cp's OEIS frontend

A160419 a(n) = A160409(n+2)/2.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, May 23 2009

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, A160418.

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

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.

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

Original entry on oeis.org

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

A160729 First differences of A160728.

Original entry on oeis.org

6, 6, 12, 24, 24, 24, 24, 24, 48, 96, 96, 48, 24, 24, 48

Offset: 1

Omar E. Pol, Jul 28 2009

Also, 6 times A160409.

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, A160161, A160407, A160409, A160728.

a(n) = A160409(n)*6.

cp's OEIS Frontend

A160419 a(n) = A160409(n+2)/2.

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

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A160408 Toothpick pyramid (see Comments lines for definition).

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

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A160729 First differences of A160728.

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