cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A172308 L-toothpick sequence in the first quadrant.

Original entry on oeis.org

0, 1, 3, 5, 7, 11, 15, 17, 21, 27, 33, 41, 47, 59, 69, 75, 79, 85, 95, 107, 117, 131, 145, 161, 177, 195, 209, 225, 237, 257, 279, 287, 295, 311, 329, 345, 355, 371, 391, 415, 441, 477, 501, 533, 563, 603, 631, 655
Offset: 0

Views

Author

Omar E. Pol, Feb 06 2010

Keywords

Comments

The same as A172310 and A172304, but starting from half L-toothpick in the first quadrant.
Note that if n is odd then we add the small L-toothpicks to the structure, otherwise we add the large L-toothpicks to the structure.
We start at stage 0 with half L-toothpick: A segment from (0,0) to (1,1).
At stage 1 we place a small L-toothpick at the exposed toothpick end.
At stage 2 we place two large L-toothpicks.
At stage 3 we place two small L-toothpicks.
At stage 4 we place two large L-toothpicks.
And so on...
The sequence gives the number of L-toothpicks after n stages. A172309 (the first differences) gives the number of L-toothpicks added at the n-th stage.

Links

Crossrefs

Cf. A139250, A153000, A160120, A160170, A160172, A161206, A161328, A172304, A172305, A172306, A172307, A172309, A172310, A172311, A172312, A172313.

Extensions

a(17)-a(47) from Robert Price, Jun 17 2019

A170876 Number of toothpicks after n stages of 3-D toothpick structure defined in Comments.

Original entry on oeis.org

0, 1, 5, 21, 37, 53, 117, 197, 261, 405, 565, 789, 965, 1221, 1541, 1941, 2453, 2933, 3621, 4389, 5093, 5909, 6805, 7925, 9093, 10629, 12197, 14133, 15733, 17717, 19493, 21605, 23909, 26453, 29109, 32117, 35013, 38085, 41285
Offset: 0

Views

Author

N. J. A. Sloane, Jan 05 2010, based on email from R. J. Mathar, Jun 02 2009 Revised by R. J. Mathar, Jan 08 2010, Jan 09 2010

Keywords

Comments

We are in 3-D, and we are placing ordinary toothpicks, as in A139250.
We start with one toothpick in the z direction
We place toothpicks at any free end, as in A139250.
We always place new toothpicks in pairs, two perpendicular toothpicks that are perpendicular to the original toothpick
The toothpicks are always in 2 out of the 3 (x, y or z) directions.
The initial values are as follows (this should be checked!):
n:.0..1..2..3..4..5..6
----------------------------
x..0..0..2..4..8..4.24 (Number added in x direction)
y..0..0..2..4..8..4.24 (Number added in y direction)
z..0..1..0..8..0..8.16 (Number added in z direction)
----------------------------
...0..1..4.16.16.16.64 (Total number added at n-th stage, A170876)
----------------------------
a..0..1..5.21.37.53.117 (Total so far, this sequence)
----------------------------

Examples

			At stage 2 we have a horizontal cross, a vertical toothpick then another horizontal cross, for a total of 5 toothpicks.
Then we add 8 vertical toothpicks at the ends of the crosses and 8 horizontal toothpicks in the same planes as the crosses, for a total of 21 toothpicks.

Links

Crossrefs

Cf. A139250, A170875 (first differences), A160160, A160170. For another version see A170837.

A160430 The 3-D toothpick sequence A160160, but using toothpicks of length 4; a(n) is the number of nodes occupied after n steps.

Original entry on oeis.org

0, 5, 13, 29, 57, 85, 113, 145, 209, 329, 509, 613, 665, 697, 761, 881, 1073, 1273, 1505, 1817, 2377, 3217, 4309, 4761, 4921, 5009, 5097, 5233, 5425, 5625, 5857, 6169, 6729, 7569, 8697, 9425, 10017, 10641, 11601, 12873
Offset: 0

Views

Author

Omar E. Pol, May 13 2009

Keywords

Examples

			Each toothpick looks like this: o-o-o-o-o.
The initial (z-axis) toothpick occupies 5 nodes. The next two, in the x-direction, add 8 further nodes, and so on.

Links

Crossrefs

Cf. A139250, A139251, A147562, A160118, A160120, A160160, A160170, A160379, A160420.

Formula

a(n) = A170884(n) + 2*A160160(n)

Extensions

Edited by N. J. A. Sloane, Jan 02 2010
Formula and more terms from Nathaniel Johnston, Nov 14 2010
Previous Showing 11-13 of 13 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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