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.

Showing 1-3 of 3 results.

A262617 First differences of A256266.

Original entry on oeis.org

0, 6, 12, 6, 24, 18, 12, 6, 48, 42, 36, 30, 24, 18, 12, 6, 96, 90, 84, 78, 72, 66, 60, 54, 48, 42, 36, 30, 24, 18, 12, 6, 192, 186, 180, 174, 168, 162, 156, 150, 144, 138, 132, 126, 120, 114, 108, 102, 96, 90, 84, 78, 72, 66, 60, 54, 48, 42, 36, 30, 24, 18, 12, 6, 384, 378, 372, 366, 360, 354, 348, 342, 336, 330, 324, 318
Offset: 0

Views

Author

Omar E. Pol, Oct 02 2015

Keywords

Comments

Number of cells turned ON at n-th stage of the cellular automaton of A256266.

Examples

			With the terms written as an irregular triangle in which the row lengths are the terms of A011782 the sequence begins:
0;
6;
12, 6;
24, 18, 12, 6;
48, 42, 36, 30, 24, 18, 12, 6;
96, 90, 84, 78, 72, 66, 60, 54, 48, 42, 36, 30, 24, 18, 12, 6;
...
Apart from the initial zero the rows list the initial terms of the positive multiples of 6 in decreasing order.

Links

Crossrefs

Cf. A008588, A011782, A080079, A255748, A256257, A256266.

Programs

  • PARI
    a(n) = if(n==0, 0, -6*n+12*2^floor(log(n)/log(2)));
vector(100, n, a(n-1)) \\ Altug Alkan, Oct 04 2015

Formula

a(n) = 6 * A080079(n), n >= 1.

A255748 Total number of ON states after n generations of cellular automaton based on triangles in a 60-degree wedge (see Comments lines for definition).

Original entry on oeis.org

1, 3, 4, 8, 11, 13, 14, 22, 29, 35, 40, 44, 47, 49, 50, 66, 81, 95, 108, 120, 131, 141, 150, 158, 165, 171, 176, 180, 183, 185, 186, 218, 249, 279, 308, 336, 363, 389, 414, 438, 461, 483, 504, 524, 543, 561, 578, 594, 609, 623, 636, 648, 659, 669, 678, 686, 693, 699, 704, 708, 711, 713, 714, 778, 841, 903, 964, 1024
Offset: 1

Views

Author

Omar E. Pol, Mar 30 2015

Keywords

Comments

Also partial sums of A080079.
In order to construct the structure we use the following rules:
On the infinite triangular grid we are in a 60-degree wedge with the vertex located on top of the wedge.
The nearest triangular cell to the vertex remains OFF.
At stage 1, we turn ON the cell whose base is adjacent to the previous OFF cell.
At stage n, in the n-th level of the structure, we turn ON k cells connected by their vertices with their bases up, where k = A080079(n).
The cells turned ON remain ON forever.
The structure seems to grow into the holes of a virtual Sierpiński's triangle (see example).
Note that this is also the structure in every one of the six wedges of the structure of A256266.
A080079 gives the number of cells turned ON at n-th stage.

Examples

			Illustration of initial terms:
-----------------------------------------------------------
n   A080079(n)   a(n)                  Diagram
-----------------------------------------------------------
.                                        / \
1       1         1                     / T \
2       2         3                    / T T \
3       1         4                   /   T   \
4       4         8                  / T T T T \
5       3        11                 /   T T T   \
6       2        13                /     T T     \
7       1        14               /       T       \
8       8        22              / T T T T T T T T \
9       7        29             /   T T T T T T T   \
10      6        35            /     T T T T T T     \
11      5        40           /       T T T T T       \
12      4        44          /         T T T T         \
13      3        47         /           T T T           \
14      2        49        /             T T             \
15      1        50       /               T               \
...
For n = 15 after 15 generations there are 50 ON cells in the structure, so a(15) = 50.

Links

Crossrefs

Cf. A047999, A001316, A080079, A139250, A169779, A169788, A170905, A233970, A256256, A256266.

Programs

  • Mathematica
    Accumulate@ Flatten@ Table[Range[2^n, 1, -1], {n, 0, 6}] (* Michael De Vlieger, Nov 03 2022 *)

Formula

a(n) = A256266(n)/6.

A262620 Number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton on the square grid (see Comments lines for definition).

Original entry on oeis.org

1, 5, 17, 21, 49, 69, 81, 85, 145, 197, 241, 277, 305, 325, 337, 341, 465, 581, 689, 789, 881, 965, 1041, 1109, 1169, 1221, 1265, 1301, 1329, 1349, 1361, 1365, 1617, 1861, 2097, 2325, 2545, 2757, 2961, 3157, 3345, 3525, 3697, 3861, 4017, 4165, 4305, 4437, 4561, 4677, 4785, 4885, 4977, 5061, 5137, 5205, 5265, 5317, 5361, 5397
Offset: 0

Views

Author

Omar E. Pol, Oct 16 2015

Keywords

Comments

On the infinite square grid consider four 90-degree wedges forming a "X" with the vertex located at the center of a cell.
At stage 0 we start with an ON cell in the vertex of the wedges, so a(0) = 1.
In order to construct the structure we use the following rules for the South wedge:
- The cells turned ON remain ON forever.
- At stage 1 we turn ON the nearest cell to the initial ON cell.
- If n is a power of 2, at stage n we turn "ON" 2*n - 1 connected cells in the n-th row of the wedge.
- Otherwise, if n is not a power of 2, at stage n we turn "ON" k - 2 connected cells in the n-th row of the wedge, where k is the number of ON cells in row n - 1.
- The "ON" cells of row n must be centered respect to the "ON" cells of row n - 1.
The structures in the other three wedges are copies of the structure in the South wedge but they grow in direction East, North and West.
Note that in every wedge the structure seems to grow into the holes of a virtual structure similar to the Sierpiński's triangle but using square cells.
A262621 gives the number of cells turned "ON" at n-th stage.
This is analog of A256266, but here we are working on the square grid and we have four wedges, not six wedges.

Examples

			Illustration of the structure after 15 generations:
.
.                                   O
.                                 O O O
.                               O O O O O
.                             O O O O O O O
.                           O O O O O O O O O
.                         O O O O O O O O O O O
.                       O O O O O O O O O O O O O
.                     O O O O O O O O O O O O O O O
.                   O               O               O
.                 O O             O O O             O O
.               O O O           O O O O O           O O O
.             O O O O         O O O O O O O         O O O O
.           O O O O O       O       O       O       O O O O O
.         O O O O O O     O O     O O O     O O     O O O O O O
.       O O O O O O O   O O O   O   O   O   O O O   O O O O O O O
.     O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O
.       O O O O O O O   O O O   O   O   O   O O O   O O O O O O O
.         O O O O O O     O O     O O O     O O     O O O O O O
.           O O O O O       O       O       O       O O O O O
.             O O O O         O O O O O O O         O O O O
.               O O O           O O O O O           O O O
.                 O O             O O O             O O
.                   O               O               O
.                     O O O O O O O O O O O O O O O
.                       O O O O O O O O O O O O O
.                         O O O O O O O O O O O
.                           O O O O O O O O O
.                             O O O O O O O
.                               O O O O O
.                                 O O O
.                                   O
.
There are 341 ON cells in the structure, so a(15) = 341.
Note that every circle in the structure should be replaced with a square cell.

Links

Crossrefs

Cf. A147562, A160720, A256266, A261692, A261693, A262621.

Formula

a(n) = 1 + 4*A261692(n).
Showing 1-3 of 3 results.

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