A182619 -id:A182619 - cp's OEIS frontend

A182617 Number of toothpicks in a toothpick spiral around n cells on hexagonal net.

0, 5, 9, 12, 15, 18, 21, 23, 26, 29, 31, 34, 36, 39, 41, 44, 46, 49, 51, 53, 56, 58, 61, 63, 65, 68, 70, 72, 75, 77, 79, 82, 84, 86, 89, 91, 93, 95, 98, 100

Offset: 0

Omar E. Pol, Dec 13 2010

The toothpick spiral contains n hexagonal "ON" cells that are connected without holes. A hexagonal cell is "ON" if the hexagon has 6 vertices that are covered by the toothpicks.

Attempt of an explanation: in the hexagonal grid, we can pick any of the hexagons as a center, and then define a ring of 6 first neighbors (hexagons adjacent to the center), then define a ring of 12 second neighbors (hexagons adjacent to any of the first ring) and so on. The current sequence describes a self-avoiding walk which starts in a spiral around the center hexagon, which covers 5 edges. The walk then takes one step to reach the rim of the first ring and travels once around this ring until it reaches a point where self-avoidance stops it. It then takes one step to reach the rim of the second ring and walks around that one, etc. Imagine that on each edge we place a toothpick if it's on the path, and interrupt counting the total number of toothpicks each time one of the hexagons has six vertices covered. The total number of toothpicks after n-th stage define this sequence. Note that, except from the last sentence, this comment is a copy from R. J. Mathar's comment in A182618 (Dec 13 2010). - Omar E. Pol, Sep 15 2013

			On the infinite hexagonal grid we start at stage 0 with no toothpicks, so a(0) = 0.
At stage 1 we place 5 toothpicks on the edges of the first hexagonal cell, so a(1) = 5.
At stage 2, from the last exposed endpoint, we place 4 other toothpicks on the edges of the second hexagonal cell, so a(2) = 5 + 4 = 9 because there are 9 toothpicks in the structure.
At stage 3, from the last exposed endpoint, we place 3 other toothpicks on the edges of the third hexagonal cell, so a(3) = 9 + 3 = 12 because there are 12 toothpicks in the spiral.
From _Omar E. Pol_, Sep 14 2013: (Start)
Illustration of initial terms:
.                        _       _         _         _
.              _       _/ \    _/ \_     _/ \_     _/ \_
.  _      _   /  _    /  _    /  _  \   /  _  \   /  _  \
. / \    / \  \ / \   \ / \   \ / \ /   \ / \ /   \ / \ /
.  _/  /  _/  /  _/   /  _/   /  _/     /  _/ \   /  _/ \
.      \_/    \_/     \_/     \_/       \_/  _/   \_/  _/
.                                                    _/
.
.  5     9      12      15       18        21       23
.
(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
Index entries for sequences related to toothpick sequences

Cf. A139250, A182618, A182619, A182632, A182840.

Conjecture: a(n) = 2*n + ceiling(sqrt(12*n - 3)), for n > 0. - Vincenzo Librandi, Sep 20 2017

A182618 Number of new grid points that are covered by the toothpicks added at n-th-stage to the toothpick spiral of A182617.

Original entry on oeis.org

6, 4, 3, 3, 3, 3, 2, 3, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 2, 3, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 3, 2, 2, 2, 2, 3

Offset: 1

Omar E. Pol, Dec 12 2010

In the toothpick spiral the toothpicks are connected by their endpoints. See A182617 for more information.
Attempt at an explanation, R. J. Mathar, Dec 13 2010: (Start)
In the hexagonal grid, we can pick any of the hexagons as a center, and then define a ring of 6 first neighbors (hexagons adjacent to the center), then define a ring of 12 second neighbors (hexagons adjacent to any of the first ring) and so on. The current sequence describes a self-avoiding walk which starts in a spiral around the center hexagon, which covers 5 edges. The walk then takes one step to reach the rim of the first ring and travels once around this ring until it reaches a point where self-avoidance stops it. It then takes one step to reach the rim of the second ring and walks around that one, etc. Imagine that on each edge we place a toothpick if it's on the path, and interrupt counting the total number of toothpicks each time one of the hexagons has six vertices covered. The first differences of these intermediate totals define the sequence. (End)

			At stage 1, starting from a node on the hexagonal net, we place 5 toothpicks on 5 edges of the first hexagon, so a(1)= 6 because there are 6 grid points that are covered by the toothpicks.
At stage 2, starting from the last exposed endpoints, we place 4 toothpicks on the edges of the second hexagon, so a(2)=4 because there are new 4 grid points that are covered by the toothpicks.
At stage 3, starting from the last exposed endpoints we place 3 toothpicks on the edges of the third hexagon, so a(3)=3 because there are new 3 grid points covered. Etc.
If written as a triangle, begins:
6,
4,3,3,3,3,2,
3,3,2,3,2,3,2,3,2,3,2,2,
3,2,3,2,2,3,2,2,3,2,2,3,2,2,3,2,2,2,
3,2,2,3,2,2,2,3,2,2,2,3,2,2,2,3,2,2,2,3,2,2,2,2,
3,2,2,2,3,2,2,2,2,3,2,2,2,2,3,2,2,2,2,3,2,2,2,2,3,2,2,2,2,2

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 toothpick sequences

Cf. A121149, A139250, A139251, A182617, A182619, A182632, A182840.

Row n has A008458(n-1) terms. Row sums give A017593.

A257481 Consider a hole-less cluster of n circles in the hexagonal lattice packing of circles; a(n) is the maximal number of circles that touch 6 circles.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 3, 3, 4

Offset: 1

Peter Woodward, Apr 26 2015

			For a(7), one circle can be completely enclosed by six surrounding circles, so a(7)=1, a(n)=0 for n<7.
For a(10), two circles can be completely enclosed by eight surrounding circles, so a(10)=2.

R. L. Graham and N. J. A. Sloane, Penny-Packing and Two-Dimensional Codes, Discrete and Comput. Geom. 5 (1990), 1-11.
Kival Ngaokrajang, Illustration of initial terms
Eric Weisstein's World of Mathematics, Hexagonal grid
Wikipedia, Circle packing

Cf. A182619, A257594, A069813.

Edited by N. J. A. Sloane, May 18 2015

cp's OEIS Frontend

A182617 Number of toothpicks in a toothpick spiral around n cells on hexagonal net.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A182618 Number of new grid points that are covered by the toothpicks added at n-th-stage to the toothpick spiral of A182617.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A257481 Consider a hole-less cluster of n circles in the hexagonal lattice packing of circles; a(n) is the maximal number of circles that touch 6 circles.

Views

Author

Keywords

Examples

Links

Crossrefs

Extensions