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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A256530 Number of ON cells at n-th stage in simple 2-dimensional cellular automaton (see Comments lines for definition).

Original entry on oeis.org

0, 1, 9, 21, 49, 61, 97, 157, 225, 237, 273, 333, 417, 525, 657, 813, 961, 973, 1009, 1069, 1153, 1261, 1393, 1549, 1729, 1933, 2161, 2413, 2689, 2989, 3313, 3661, 3969, 3981, 4017, 4077, 4161, 4269, 4401, 4557, 4737, 4941, 5169, 5421, 5697, 5997, 6321, 6669, 7041, 7437, 7857, 8301, 8769, 9261, 9777, 10317, 10881, 11469
Offset: 0

Views

Author

Omar E. Pol, Apr 21 2015

Keywords

Comments

On the infinite square grid at stage 0 there are no ON cells, so a(0) = 0.
At stage 1, only one cell is turned ON, so a(1) = 1.
If n is a power of 2 so the structure is a square of side length 2n - 1 that contains (2n-1)^2 ON cells.
The structure grows by the four corners as square waves forming layers of ON cells up the next square structure, and so on (see example).
Note that a(24) = 1729 is also the Hardy-Ramanujan number (see A001235).
Has the same rules as A256534 but here a(1) = 1 not 4.
Has a smoother behavior than A160414 with which shares infinitely many terms (see example).
A256531, the first differences, gives the number of cells turned ON at n-th stage.

Examples

			With the positive terms written as an irregular triangle in which the row lengths are the terms of A011782 the sequence begins:
1;
9;
21,    49;
61,    97,  157,  225;
237,  273,  333,  417,  525,  657,  813,  961;
...
Right border gives A060867.
This triangle T(n,k) shares with the triangle A160414 the terms of the column k, if k is a power of 2, for example both triangles share the following terms: 1, 9, 21, 49, 61, 97, 225, 237, 273, 417, 961, etc.
.
Illustration of initial terms, for n = 1..10:
.       _ _ _ _                       _ _ _ _
.      |  _ _  |                     |  _ _  |
.      | |  _|_|_ _ _ _ _ _ _ _ _ _ _|_|_  | |
.      | |_|  _ _ _ _ _ _   _ _ _ _ _ _  |_| |
.      |_ _| |  _ _ _ _  | |  _ _ _ _  | |_ _|
.          | | |  _ _  | | | |  _ _  | | |
.          | | | |  _|_|_|_|_|_|_  | | | |
.          | | | |_|  _ _   _ _  |_| | | |
.          | | |_ _| |  _|_|_  | |_ _| | |
.          | |_ _ _| |_|  _  |_| |_ _ _| |
.          |  _ _ _|  _| |_| |_  |_ _ _  |
.          | |  _ _| | |_ _ _| | |_ _  | |
.          | | |  _| |_ _| |_ _| |_  | | |
.          | | | | |_ _ _ _ _ _ _| | | | |
.          | | | |_ _| | | | | |_ _| | | |
.       _ _| | |_ _ _ _| | | |_ _ _ _| | |_ _
.      |  _| |_ _ _ _ _ _| |_ _ _ _ _ _| |_  |
.      | | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | |
.      | |_ _| |                     | |_ _| |
.      |_ _ _ _|                     |_ _ _ _|
.
After 10 generations there are 273 ON cells, so a(10) = 273.

Links

Crossrefs

Cf. A000225, A011782, A001235, A060867, A139250, A147562, A160410, A160414, A256531, A256534.

Programs

  • Mathematica
    With[{z=7},Join[{0},Flatten[Array[(2^#-1)^2+12Range[0,2^(#-1)-1]^2&,z]]]] (* Generates 2^z terms *) (* Paolo Xausa, Nov 15 2023, after Omar E. Pol *)

Formula

For i = 1 to z: for j = 0 to 2^(i-1)-1: n = n+1: a(n) = (2^i-1)^2 + 3*(2*j)^2: next j: next i

A256534 Number of ON cells at n-th stage in simple 2-dimensional cellular automaton (see Comments lines for definition).

Original entry on oeis.org

0, 4, 16, 28, 64, 76, 112, 172, 256, 268, 304, 364, 448, 556, 688, 844, 1024, 1036, 1072, 1132, 1216, 1324, 1456, 1612, 1792, 1996, 2224, 2476, 2752, 3052, 3376, 3724, 4096, 4108, 4144, 4204, 4288, 4396, 4528, 4684, 4864, 5068, 5296, 5548, 5824, 6124, 6448, 6796, 7168, 7564, 7984, 8428, 8896, 9388, 9904, 10444, 11008
Offset: 0

Views

Author

Omar E. Pol, Apr 22 2015

Keywords

Comments

On the infinite square grid at stage 0 there are no ON cells, so a(0) = 0.
At stage 1, four cells are turned ON forming a square, so a(1) = 4.
If n is a power of 2 so the structure is a square of side length 2n that contains (2n)^2 ON cells.
The structure grows by the four corners as square waves forming layers of ON cells up the next square structure, and so on (see example).
Has the same rules as A256530 but here a(1) = 4 not 1.
Has a smoother behavior than A160410 with which shares infinitely many terms (see example).
A261695, the first differences, gives the number of cells turned ON at n-th stage.

Examples

			With the positive terms written as an irregular triangle in which the row lengths are the terms of A011782 the sequence begins:
4;
16;
28,     64;
76,    112,  172,  256;
268,   304,  364,  448,  556,  688,  844, 1024;
...
Right border gives the elements of A000302 greater than 1.
This triangle T(n,k) shares with the triangle A160410 the terms of the column k, if k is a power of 2, for example, both triangles share the following terms: 4, 16, 28, 64, 76, 112, 256, 268, 304, 448, 1024, etc.
.
Illustration of initial terms, for n = 1..10:
.       _ _ _ _                         _ _ _ _
.      |  _ _  |                       |  _ _  |
.      | |  _|_|_ _ _ _ _ _ _ _ _ _ _ _|_|_  | |
.      | |_|  _ _ _ _ _ _     _ _ _ _ _ _  |_| |
.      |_ _| |  _ _ _ _  |   |  _ _ _ _  | |_ _|
.          | | |  _ _  | |   | |  _ _  | | |
.          | | | |  _|_|_|_ _|_|_|_  | | | |
.          | | | |_|  _ _     _ _  |_| | | |
.          | | |_ _| |  _|_ _|_  | |_ _| | |
.          | |_ _ _| |_|  _ _  |_| |_ _ _| |
.          |       |   | |   | |   |       |
.          |  _ _ _|  _| |_ _| |_  |_ _ _  |
.          | |  _ _| | |_ _ _ _| | |_ _  | |
.          | | |  _| |_ _|   |_ _| |_  | | |
.          | | | | |_ _ _ _ _ _ _ _| | | | |
.          | | | |_ _| | |   | | |_ _| | | |
.       _ _| | |_ _ _ _| |   | |_ _ _ _| | |_ _
.      |  _| |_ _ _ _ _ _|   |_ _ _ _ _ _| |_  |
.      | | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | |
.      | |_ _| |                       | |_ _| |
.      |_ _ _ _|                       |_ _ _ _|
.
After 10 generations there are 304 ON cells, so a(10) = 304.

Links

Crossrefs

Cf. A000302, A011782, A139250, A147562, A160410, A160414, A236305, A256530, A261695.

Programs

  • Mathematica
    {0}~Join~Flatten@ Table[4^i + 3 (2 j)^2, {i, 6}, {j, 0, 2^(i - 1) - 1}] (* Michael De Vlieger, Nov 03 2022 *)

Formula

For i = 1 to z: for j = 0 to 2^(i-1)-1: n = n+1: a(n) = 4^i + 3*(2*j)^2: next j: next i
It appears that a(n) = 4 * A236305(n-1), n >= 1.

A160727 a(n) = A161415(n+1)/4.

Original entry on oeis.org

2, 3, 7, 3, 9, 9, 23, 3, 9, 9, 27, 9, 27, 27, 73, 3, 9, 9, 27, 9, 27, 27, 81, 9, 27, 27, 81, 27, 81, 81, 227, 3, 9, 9, 27, 9, 27, 27, 81, 9, 27, 27, 81, 27, 81, 81, 243, 9, 27, 27, 81, 27, 81, 81, 243, 27, 81, 81, 243, 81, 243, 243, 697, 3, 9, 9, 27, 9
Offset: 1

Views

Author

Omar E. Pol, Jun 13 2009

Keywords

Examples

			From _Omar E. Pol_, Jan 01 2014: (Start)
Written as an irregular triangle in which row lengths is A000079 the sequence begins:
2;
3,7;
3,9,9,23;
3,9,9,27,9,27,27,73;
3,9,9,27,9,27,27,81,9,27,27,81,27,81,81,227;
3,9,9,27,9,27,27,81,9,27,27,81,27,81,81,243,9,27,27,81,27, 81,81,243,27,81,81,243,81,243,243,697;
(End)

Links

Crossrefs

Cf. A152978, A153000, A160414, A161415.

Programs

  • Mathematica
    A160727[n_]:=3^DigitCount[n,2,1]-If[IntegerQ[Log2[n+1]],(n+1)/2,0];Array[A160727,100] (* Paolo Xausa, Sep 01 2023 *)

Formula

a(n) = A048883(n), except a(n) = A048883(n) - (n+1)/2 if n is a power of 2 minus 1. - Omar E. Pol, Jan 06 2014

Extensions

a(11)-a(58) from M. F. Hasler, Dec 03 2012
a(59)-a(68) from Omar E. Pol, Jan 06 2014

A350632 a(n) is the total number of ON cells at stage n of a cellular automaton where cells are turned ON when they have one or two neighbors ON (see Comments for precise definition).

Original entry on oeis.org

0, 1, 9, 21, 45, 57, 85, 121, 177, 189, 217, 253, 329, 373, 465, 557, 721, 737, 765, 801, 877, 921, 1013, 1105, 1301, 1377, 1485, 1601, 1805, 1985, 2221, 2449, 2873, 2909, 2937, 2973, 3049, 3093, 3185, 3277, 3473, 3549, 3657, 3773, 3977, 4157, 4393, 4621, 5113
Offset: 0

Views

Author

Rémy Sigrist, Jan 08 2022

Keywords

Comments

On the infinite square grid, start with all cells OFF.
Turn a single cell to the ON state.
At each subsequent step, each cell with exactly one or two neighbors ON is turned ON, and everything that is already ON remains ON.
Here "neighbor" refers to the eight adjacent and diagonally adjacent cells in the Moore neighborhood.

Examples

			The first 5 generations can be depicted as follows:
         . . . . . . . . . . .
         . 5 5 . . . . . 5 5 .
         . 5 4 4 4 4 4 4 4 5 .
         . . 4 3 3 . 3 3 4 . .
         . . 4 3 2 2 2 3 4 . .
         . . 4 . 2 1 2 . 4 . .
         . . 4 3 2 2 2 3 4 . .
         . . 4 3 3 . 3 3 4 . .
         . 5 4 4 4 4 4 4 4 5 .
         . 5 5 . . . . . 5 5 .
         . . . . . . . . . . .
- so a(0) = 0,
     a(1) = 0 + 1 = 1,
     a(2) = 1 + 8 = 9,
     a(3) = 9 + 12 = 21,
     a(4) = 21 + 24 = 45,
     a(5) = 45 + 12 = 57.

Links

Crossrefs

Cf. A147562, A151725, A160414, A256530, A350633, A350639.

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]

A162349 First differences of A160412.

Original entry on oeis.org

3, 9, 9, 27, 9, 27, 27, 81, 9, 27, 27, 81, 27, 81, 81, 243, 9, 27, 27, 81, 27, 81, 81, 243, 27, 81, 81, 243, 81, 243, 243, 729, 9, 27, 27, 81, 27, 81, 81, 243, 27, 81, 81, 243, 81, 243, 243, 729, 27, 81, 81, 243, 81, 243, 243, 729, 81, 243, 243, 729, 243, 729
Offset: 1

Views

Author

Omar E. Pol, Jul 14 2009

Keywords

Comments

Note that if A048883 is written as a triangle then rows converge to this sequence. - Omar E. Pol, Nov 15 2009

Links

Crossrefs

Cf. A063787, A152980, A160410, A160411, A160412, A160414, A160415, A160417, A160797.
Cf. A048883, A161415. - Omar E. Pol, Nov 15 2009

Programs

  • Mathematica
    a[n_] := 3^(1 + DigitCount[n - 1, 2, 1]); Array[a, 100] (* Amiram Eldar, Feb 02 2024 *)

Formula

a(n) = 3^A063787(n) = 3 * A048883(n-1). - Amiram Eldar, Feb 02 2024

Extensions

More terms from Omar E. Pol, Nov 15 2009
More terms from Colin Barker, Apr 19 2015
More terms from Amiram Eldar, Feb 02 2024

A161341 First differences of A161340.

Original entry on oeis.org

1, 26, 56, 260, 56, 392, 392, 2192, 56, 392, 392, 2744, 392, 2744, 2744, 16952, 56, 392, 392, 2744, 392, 2744, 2744, 19208, 392, 2744, 2744, 19208, 2744, 19208, 19208, 125336, 56, 392, 392, 2744, 392, 2744, 2744, 19208, 392, 2744, 2744, 19208, 2744, 19208, 19208
Offset: 1

Views

Author

Omar E. Pol, Jun 14 2009

Keywords

Examples

			From _Omar E. Pol_, Mar 15 2020: (Start)
Written as an irregular triangle in which row lengths give A011782 the sequence begins:
1;
26;
56, 260;
56, 392, 392, 2192;
56, 392, 392, 2744, 392, 2744, 2744, 16952;
56, 392, 392, 2744, 392, 2744, 2744, 19208, 392, 2744, 2744, 19208, 2744, ...
(End)

Crossrefs

Cf. A011782, A160414, A161340, A161415.

Programs

  • PARI
    f(n) = 8*7^hammingweight(n-1); \\ A160429
ispow2(n) = my(k); (n==2) || (ispower(n,,&k) && (k==2));
a(n) = if (n==1, 1, if (ispow2(n), f(n) - 3*n*(3*n - 1), f(n))); \\ Michel Marcus, Mar 15 2020

Formula

a(n) = A160429(n) for n>1 and n not a power of 2.
a(n) = A160429(n) - 3n*(3n - 1) for n>1 and n a power of 2.

Extensions

Formula and more terms from Nathaniel Johnston, Nov 15 2010
More terms from Jinyuan Wang, Mar 14 2020

A262609 Divisors of 1728.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, 48, 54, 64, 72, 96, 108, 144, 192, 216, 288, 432, 576, 864, 1728
Offset: 1

Views

Author

Omar E. Pol, Nov 20 2015

Keywords

Comments

A000578(12) = 1728 is the cube of 12.
The number of divisors of 1728 is A000005(1728) = 28.
The sum of the divisors of 1728 is A000203(1728) = 5080.
The prime factorization of 1728 is 2^6 * 3^3.
1728 + 1 = A001235(1) = A011541(2) = 1729 is the Hardy-Ramanujan number.
Three examples related to cellular automata:
1728 is also the number of ON cells after 32 generations of the cellular automata A160239 and A253088.
1728 is also the total number of ON cells around the central ON cell after 24 generations of the cellular automata A160414 and A256530.
1728 is also the total number of ON cells around the central ON cell after 43 generations of the cellular automata A160172 and A255366.

Examples

			a(3) * a(26) = 3 * 576 = 1728.
a(4) * a(25) = 4 * 432 = 1728.
a(5) * a(24) = 6 * 288 = 1728.

Links

Crossrefs

Cf. A000005, A000203, A000578, A001235, A011541, A133029, A160172, A160239, A160414, A246030, A253088, A255366, A256530.

Programs

  • Mathematica
    Divisors[1728]
  • PARI
    divisors(1728)
  • Sage
    divisors(1728);
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