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 1-6 of 6 results.

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

A160118 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, 309, 393, 429, 681, 685, 713, 725, 809, 821, 905, 941, 1193, 1205, 1289, 1325, 1577, 1613, 1865, 1973, 2729, 2733, 2761, 2773, 2857, 2869, 2953, 2989, 3241, 3253, 3337, 3373, 3625, 3661, 3913, 4021, 4777, 4789
Offset: 0

Views

Author

Omar E. Pol, May 05 2009

Keywords

Comments

On the infinite square grid, we start at stage 0 with all square cells in the OFF state.
Define a "peninsula cell" to a cell that is connected to the structure by exactly one of its vertices.
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 in ON at stage n-1.
For n>1, if n is odd, at stage n we turn ON all the peninsular OFF cells.
For the corresponding corner sequence, see A160796.
An animation will show the fractal-like behavior (cf. A139250).
For the first differences see A160415. - Omar E. Pol, Mar 21 2011
First differs from A188343 at a(13). - Omar E. Pol, Mar 28 2011

Examples

			If we label the generations of cells turned ON by consecutive numbers we get the cell pattern shown below:
9...............9
.888.888.888.888.
.878.878.878.878.
.8866688.8866688.
...656.....656...
.8866444.4446688.
.878.434.434.878.
.888.4422244.888.
.......212.......
.888.4422244.888.
.878.434.434.878.
.8866444.4446688.
...656.....656...
.8866688.8866688.
.878.878.878.878.
.888.888.888.888.
9...............9
In the first generation, only the central "1" is ON, a(1)=1. In the next generation, we turn ON eight "2" around the central cell, leading to a(2)=a(1)+8=9. In the third generation, four "3" are turned ON at the vertices of the square, a(3)=a(2)+4=13. And so on...

Links

Crossrefs

Cf. A139250, A139251, A147562, A147610, A160117, A160119, A160379, A160415, A160796, A188343.

Programs

  • Mathematica
    With[{d = 2}, wt[n_] := DigitCount[n, 2, 1]; a[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}]]; a[0] = 0; a[1] = 1; Array[a, 50, 0]] (* Amiram Eldar, Aug 01 2023 *)

Formula

From Nathaniel Johnston, Mar 24 2011: (Start)
a(2n-1) = 9 + 4*Sum_{k=2..n} A147610(k) + 28*Sum_{k=2..n-1} A147610(k), n >= 2.
a(2n) = 9 + 4*Sum_{k=2..n} A147610(k) + 28*Sum_{k=2..n} A147610(k), n >= 1.
(End)

Extensions

Entry revised by Omar E. Pol and N. J. A. Sloane, Feb 16 2010, Feb 21 2010
a(8) - a(38) from Nathaniel Johnston, Nov 06 2010
a(13) corrected at the suggestion of Sean A. Irvine. Then I corrected 19 terms between a(14) and a(38). Finally I added a(39)-a(42). - Omar E. Pol, Mar 21 2011
Rule, for n even, edited by Omar E. Pol, Mar 22 2011
Incorrect comment (in "formula" section) removed by Omar E. Pol, Mar 23 2011, with agreement of author.
More terms from Amiram Eldar, Aug 01 2023

A160117 Number of "ON" cells after n-th stage in simple 2-dimensional cellular automaton (see Comments for precise definition).

Original entry on oeis.org

0, 1, 9, 13, 41, 49, 101, 113, 189, 205, 305, 325, 449, 473, 621, 649, 821, 853, 1049, 1085, 1305, 1345, 1589, 1633, 1901, 1949, 2241, 2293, 2609, 2665, 3005, 3065, 3429, 3493, 3881, 3949, 4361, 4433, 4869, 4945, 5405, 5485, 5969, 6053, 6561, 6649, 7181, 7273
Offset: 0

Views

Author

Omar E. Pol, May 05 2009, May 15 2009

Keywords

Comments

Define "peninsula cell" to be the "ON" cell connected to the structure by exactly one of its vertices.
Define "bridge cell" to be the "ON" cell connected to two cells of the structure by exactly consecutive two of its vertices.
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 even, we turn "ON" the cells around the cells turned "ON" at the generation n-1.
- If n is odd, we turn "ON" the possible bridge cells and the possible peninsula cells.
- Everything that is already ON remains ON.
A160411, the first differences, gives the number of cells turned "ON" at n-th stage.

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
.886644222446688.
.878.434.434.878.
.886644464446688.
9..656.656.656..9
.886668666866688.
.878.878.878.878.
.888.888.888.888.
9...9...9...9...9
At the first generation, only the central "1" is ON, so a(1) = 1. At the second generation, we turn ON eight cells around the central cell, leading to a(2) = a(1)+8 = 9. At the third generation, we turn ON four peninsula cells, so a(3) = a(2)+4 = 13. At the fourth generation, we turn ON the cells around the cells turned ON at the third generation, so a(4) = a(3)+28 = 41. At the 5th generation, we turn ON four peninsula cells and four bridge cells, so a(5) = a(4)+8 = 49.

Links

Crossrefs

Cf. A139250, A139251, A147562, A160118, A160119, A160379.

Programs

  • Maple
    a:= proc(n) local r;
      r:= irem(n, 2);
      `if`(n<2, n, 5+(n-r)*((7*n-3*r)/2-5))
    end:
seq(a(n), n=0..80);  # Alois P. Heinz, Sep 16 2011
  • Mathematica
    a[0] = 0; a[1] = 1; a[n_] := If[EvenQ[n], (7n^2 - 10n + 10)/2, (7n^2 - 20n + 23)/2]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Jul 16 2015, after Nathaniel Johnston *)

Formula

a(2n) = 5 + 2n(7n-5) for n>=1, a(2n+1) = 5 + 2n(7n-3) for n>=1. - Nathaniel Johnston, Nov 06 2010
G.f.: x*(x^2+1)*(4*x^3+x^2+8*x+1)/((x+1)^2*(1-x)^3). - Alois P. Heinz, Sep 16 2011

Extensions

a(10) - a(27) from Nathaniel Johnston, Nov 06 2010
a(28) - a(47) from Alois P. Heinz, Sep 16 2011

A160428 Number of ON cells at n-th stage of three-dimensional version of the cellular automaton A160410, using cubes.

Original entry on oeis.org

0, 8, 64, 120, 512, 568, 960, 1352, 4096, 4152, 4544, 4936, 7680, 8072, 10816, 13560, 32768, 32824, 33216, 33608, 36352, 36744, 39488, 42232, 61440, 61832, 64576, 67320, 86528, 89272, 108480, 127688, 262144, 262200, 262592, 262984, 265728, 266120, 268864, 271608
Offset: 0

Views

Author

Omar E. Pol, Jun 01 2009

Keywords

Links

Crossrefs

Cf. A139250, A160119, 160379, A160410, A160429, A161340, A161342.

Programs

  • Mathematica
    a[n_] := 8*Sum[7^DigitCount[k, 2, 1], {k, 0, n - 1}]; Array[a, 40, 0] (* Michael De Vlieger, Nov 01 2022 *)

Formula

a(n) = 8 * Sum_{k=0..n-1} 7^A000120(k)
a(n) = 8 + 56 * Sum_{k=1..n-1} A151785(k) for n >= 1

Extensions

Formulas and more terms from Nathaniel Johnston, Nov 13 2010
More terms from Michael De Vlieger, Nov 01 2022

A161340 Number of "ON" cells at n-th stage of three-dimensional version of the cellular automaton A160414 using cubes.

Original entry on oeis.org

0, 1, 27, 83, 343, 399, 791, 1183, 3375, 3431, 3823, 4215, 6959, 7351, 10095, 12839, 29791, 29847, 30239, 30631, 33375, 33767, 36511, 39255, 58463, 58855, 61599, 64343, 83551, 86295, 105503, 124711, 250047, 250103, 250495, 250887, 253631, 254023, 256767, 259511
Offset: 0

Views

Author

Omar E. Pol, Jun 14 2009

Keywords

Comments

For the first differences see A161341, where an explicit formula is given.

Links

Crossrefs

Cf. A000079, A000578, A139250, A160119, A160379, A160414, A160428, A161341.

Formula

a(n) = (2n-1)^3 if n is a power of 2.

Extensions

Edited by Omar E. Pol, Sep 05 2009, Nov 21 2010
More terms from Nathaniel Johnston, Nov 15 2010
More terms from Jinyuan Wang, Mar 14 2020

A163989 First differences of A160379.

Original entry on oeis.org

1, 26, 8, 200, 26, 620, 56, 1304, 98, 2252, 152, 3464, 218, 4940, 296, 6680, 386, 8684, 488, 10952, 602, 13484, 728, 16280, 866, 19340, 1016, 22664, 1178, 26252, 1352, 30104, 1538, 34220, 1736, 38600, 1946, 43244, 2168, 48152, 2402, 53324, 2648, 58760, 2906
Offset: 1

Views

Author

Omar E. Pol, Sep 20 2009

Keywords

Links

Crossrefs

Cf. A160117, A160119, A160379, A160428, A161340, A163987.

Programs

  • Mathematica
    LinearRecurrence[{0,3,0,-3,0,1},{1,26,8,200,26,620,56,1304},50] (* Harvey P. Dale, May 19 2025 *)
  • PARI
    Vec(-x*(18*x^3+x^2+26*x+1)*(x^4+4*x^2+1)/((x-1)^3*(x+1)^3)  + O(x^100)) \\ Colin Barker, Sep 17 2013

Formula

a(2n+1) = 6n^2 + 2 for n>=1.
a(2n) = 132n^2 - 240n + 152 for n>=2.
From Colin Barker, Sep 17 2013: (Start)
a(n) = 3*a(n-2)-3*a(n-4)+a(n-6) for n>8.
G.f.: -x*(18*x^3+x^2+26*x+1)*(x^4+4*x^2+1) / ((x-1)^3*(x+1)^3). (End)

Extensions

Formula and more terms from Nathaniel Johnston, Nov 15 2010
More terms from Colin Barker, Sep 17 2013
Showing 1-6 of 6 results.

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