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.

Previous Showing 11-17 of 17 results.

A160417 a(n) = A160415(n+1)/4.

Original entry on oeis.org

2, 1, 7, 1, 7, 3, 21, 1, 7, 3, 21, 3, 21, 9, 63, 1, 7, 3, 21, 3, 21, 9, 63, 3, 21, 9, 63, 9, 63, 27, 189, 1, 7, 3, 21, 3, 21, 9, 63, 3, 21, 9, 63, 9, 63, 27, 189, 3, 21, 9, 63, 9, 63, 27, 189, 9, 63, 27, 189, 27, 189, 81, 567, 1, 7, 3, 21, 3, 21, 9, 63, 3, 21
Offset: 1

Views

Author

Omar E. Pol, Jun 13 2009

Keywords

Crossrefs

Cf. A152978, A160118, A160413, A160415.

Programs

  • Mathematica
    With[{d = 2}, wt[n_] := DigitCount[n, 2, 1]; f[n_] := If[OddQ[n], 3^d + (2^d) * Sum[(2^d - 1)^(wt[k] - 1), {k, 1, (n - 1)/2}] + (2^d)*(3^d - 2) * Sum[(2^d - 1)^(wt[k] - 1), {k, 1, (n - 3)/2}], 3^d + (2^d) * Sum[(2^d - 1)^(wt[k] - 1), {k, 1, n/2 - 1}] + (2^d)*(3^d - 2) * Sum[(2^d - 1)^(wt[k] - 1), {k, 1, n/2 - 1}]]; f[0] = 0; f[1] = 1; Differences[Array[f, 100]]/4] (* Amiram Eldar, Feb 02 2024 *)

Extensions

More terms from Max Alekseyev, Dec 12 2011
More terms from Amiram Eldar, Feb 02 2024

A173456 Number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton (see Comments for precise definition).

Original entry on oeis.org

0, 1, 9, 21, 25, 53, 89, 93, 121, 157, 169, 253, 361, 365, 393, 429, 441, 525, 633, 645, 729, 837, 873, 1125, 1449, 1453, 1481, 1517, 1529, 1613, 1721, 1733, 1817, 1925, 1961, 2213, 2537, 2549, 2633, 2741, 2777, 3029, 3353, 3389, 3641, 3965, 4073, 4829, 5801
Offset: 0

Views

Author

Omar E. Pol, Feb 18 2010

Keywords

Comments

On the infinite square grid, we start at stage 0 with all cells in OFF state. At stage 1, we turn ON a single cell, in the central position.
In order to construct this sequence we use the following rules:
- If n is congruent to 0 (mod 3), we turn "ON" the cells around the vertex of every convex corner formed in the structure at the generation n-1. Note that every vertex is surrounded by three new "ON" cells.
- If n is congruent to 1 (mod 3), we turn "ON" the possible peninsula cells (For the definition of peninsula cell see A160117).
- If n is congruent to 2 (mod 3), we turn "ON" the cells around the cells turned "ON" at the generation n-1.
- Everything that is already ON remains ON.
A173457, the first differences, gives the number of cells turned "ON" at n-th stage.

Examples

			Array begins:
0, 1, 9;
21, 25, 53;
89, 93, 121;
157, 169, 253;
361, 365, 393;
...
If we label the generations of cells turned ON by consecutive numbers we get the cell pattern shown below:
7...........7
.66.66.66.66.
.65556.65556.
..545...545..
.65533.33556.
.66.32223.66.
.....212.....
.66.32223.66.
.65533.33556.
..545...545..
.65556.65556.
.66.66.66.66.
7...........7

Links

Crossrefs

Cf. A160117, A160118, A173457, A173458, A173459, A173460.

Formula

a(0)=0, a(n)=a(n-1)+A173457(n), n>=1

Extensions

a(41)-a(48) from Lars Blomberg, Apr 22 2013

A173460 Number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton (see Comments for precise definition).

Original entry on oeis.org

0, 1, 9, 21, 29, 81, 93, 105, 189, 225, 253, 441, 453, 465, 549, 585, 621, 873, 909, 945, 1197, 1305, 1397, 2025, 2037, 2049, 2133, 2169, 2205, 2457, 2493, 2529, 2781, 2889, 2997, 3753, 3789, 3825, 4077, 4185, 4293, 5049, 5157, 5265, 6021, 6345, 6637, 8649
Offset: 0

Views

Author

Omar E. Pol, Feb 18 2010

Keywords

Comments

On the infinite square grid, we start at stage 0 with all cells in OFF state. At stage 1, we turn ON a single cell, in the central position.
In order to construct this sequence we use the following rules:
- If n is congruent to 0 (mod 3), we turn "ON" the cells around the vertex of every convex corner formed in the structure at the generation n-1. Note that every vertex is surrounded by three new "ON" cells.
- If n is congruent to 1 (mod 3), we turn "ON" the possible bridge cells and the possible peninsula cells (For the definition of bridge cell and of peninsula cell see A160117).
- If n is congruent to 2 (mod 3), we turn "ON" the cells around the cells turned "ON" at the generation n-1.
- Everything that is already ON remains ON.
A173461, the first differences, gives the number of cells turned "ON" at n-th stage.

Examples

			Array begins:
0, 1, 9;
21, 29, 81;
93, 105, 189;
225, 253, 441;
453, 465, 549;
...
If we label the generations of cells turned ON by consecutive numbers we get the cell pattern shown below:
7..7.....7..7
.66.......66.
.65555555556.
7.545545545.7
..553353355..
..553222355..
..545212545..
..553222355..
..553353355..
7.545545545.7
.65555555556.
.66.......66.
7..7.....7..7

Links

Crossrefs

Cf. A160117, A160118, A173456, A173461, A173462, A173463.

Formula

a(0)=0, a(n) = a(n-1) + A173461(n), n>=1. - [Lars Blomberg, Apr 23 2013]

Extensions

a(18)-a(47) from Lars Blomberg, Apr 23 2013

A160416 Number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton (see Comments for precise definition).

Original entry on oeis.org

0, 1, 8, 11, 32, 39, 80, 89, 146, 159
Offset: 0

Views

Author

Omar E. Pol, May 20 2009, Jun 14 2009

Keywords

Examples

			If we label the generations of cells turned ON by consecutive numbers we get the cell pattern shown below:
..9...9...9...9...9
...888.888.888.888.
...878.878.878.878.
...886668666866688.
..9..656.656.656..9
...886644464446688.
...878.434.434.878.
...886644222446688.
..9..656.212.656..9
000000000022446688.
0000000000.434.878.
000000000064446688.
000000000056.656..9
000000000066866688.
0000000000.878.878.
0000000000.888.888.
00000000009...9...9
0000000000.........
0000000000.........

Links

Crossrefs

Cf. A139250, A139251, A152980, A153006, A160117, A160118, A160410, A160412, A160414.
Cf. A160796, A161417. [From Omar E. Pol, Jun 14 2009]

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

A188343 Number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton (see Comments for precise definition).

Original entry on oeis.org

0, 1, 9, 13, 41, 45, 73, 85, 169, 173, 201, 213, 297, 317, 441, 485, 681, 685, 713, 725, 809, 829, 953, 997, 1193, 1213, 1353, 1397, 1689, 1773, 2201, 2309, 2737, 2765, 2817, 2845, 2953, 2989, 3113
Offset: 0

Views

Author

Nathaniel Johnston, Mar 28 2011

Keywords

Comments

On the infinite square grid, we start at stage 0 with all square cells in the OFF state.
At stage 1 we turn ON a single cell in the central position.
For n>1, if n is even, at stage n we turn ON all the OFF neighboring cells from cells that were turned ON at stage n-1.
For n>1, if n is odd, at stage n we turn ON all the OFF cells that have exactly one neighbor that was turned ON at stage n-1.
First differs from A160118 at a(13).

Links

Crossrefs

Cf. A160118.

Formula

For n even, use Generations rule "/12345678/250". For n odd, use Generations rule "/1/250".

A163987 First differences of A160119.

Original entry on oeis.org

1, 26, 8, 200, 8, 200, 56, 1400, 8, 200, 56, 1400, 56, 1400, 392, 9800, 8, 200, 56, 1400, 56, 1400, 392, 9800, 56, 1400, 392, 9800, 392, 9800, 2744, 68600
Offset: 1

Views

Author

Omar E. Pol, Sep 20 2009

Keywords

Links

Crossrefs

Cf. A160118, A160119, A160379, A160428, A161340, A163989.

Formula

a(2n-1) = 8*A151785(n-1), n >= 2, a(2n) = 200*A151785(n-1), n >= 2. - Nathaniel Johnston, Mar 24 2011

Extensions

a(8)-a(32) from Nathaniel Johnston, Mar 24 2011
Previous Showing 11-17 of 17 results.

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

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