A168113 -id:A168113 - cp's OEIS frontend

A168114 If A168113 is regarded as a triangle then the rows converge to this sequence.

3, 6, 7, 8, 9, 16, 17, 12, 9, 16, 19, 22, 31, 46, 41, 20, 9, 16, 19, 22, 31, 46, 43, 30, 31, 48, 57, 72, 105, 130, 97, 36, 9, 16, 19, 22, 31, 46, 43, 30, 31, 48, 57, 72, 105, 130, 99, 46, 31, 48, 57, 72, 105, 132, 113, 88, 107, 150, 183, 246, 337, 354, 225

Offset: 1

Omar E. Pol, Dec 07 2009

nonn

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, A147646, A168112, A168113.

Terms after a(15) from Nathaniel Johnston, Mar 28 2011

A160731 First differences of A160730.

Original entry on oeis.org

2, 2, 4, 6, 6, 6, 12, 14, 12, 6, 12, 14, 16, 18, 32, 34, 20, 6, 12, 14, 16, 18, 32, 34, 24, 18, 32, 38, 44, 62, 92, 82, 36, 6, 12, 14, 16, 18, 32, 34, 24, 18, 32, 38, 44, 62, 92, 82, 40, 18, 32, 38, 44, 62

Offset: 1

Omar E. Pol, May 25 2009

nonn

			Placing the entries starting from a(4) in a triangle with rows that have length equal to powers of two gives:
6, 6
6, 12, 14, 12
6, 12, 14, 16, 18, 32, 34, 20
6, 12, 14, 16, 18, 32, 34, 24, 18, 32, 38, 44, 62, 92, 82, 36
...
The rows of this triangle tend to 2*A168114.

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, A160730, A160733, A160737, A160739, A160741.

From Nathaniel Johnston, Mar 28 2011: (Start)
a(n) = 2*A168113(n)
a(2^(n+2) + 1) = 4(2^n + 1), n >= 1.
(End)

Terms after a(11) from Nathaniel Johnston, Mar 28 2011

A168112 Toothpick sequence starting with a straight line, with angle = Pi/4, from which protrudes a half toothpick.

Original entry on oeis.org

0, 1, 2, 4, 7, 10, 13, 19, 26, 32, 35, 41, 48, 56, 65, 81, 98, 108, 111, 117, 124, 132, 141, 157, 174, 186, 195, 211, 230, 252, 283, 329, 370, 388, 391, 397, 404, 412, 421, 437, 454, 466, 475, 491, 510, 532, 563, 609, 650, 670

Offset: 0

Omar E. Pol, Dec 07 2009

nonn

On the infinite square grid, we start at round 0 drawing a straight line, with angle = Pi/4, from which protrudes a half toothpick.
At round 1 we place an orthogonal toothpick centered at the end.
In each subsequent round, for every exposed toothpick end, place an orthogonal toothpick centered at that end.
The sequence gives the number of toothpicks after n rounds.
See also A168113, the first differences.
For more information see A139250, which is the main entry for this sequence.

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, A153000, A153006, A160406, A168113, A168114.

a(n) = A160730(n)/2. [From Nathaniel Johnston, Mar 28 2011]

Terms after a(34) from Nathaniel Johnston, Mar 28 2011

cp's OEIS Frontend

A168114 If A168113 is regarded as a triangle then the rows converge to this sequence.

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A160731 First differences of A160730.

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A168112 Toothpick sequence starting with a straight line, with angle = Pi/4, from which protrudes a half toothpick.

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Extensions