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 1-5 of 5 results.

A161342 Number of "ON" cubic cells at n-th stage in simple 3-dimensional cellular automaton: a(n) = A160428(n)/8.

Original entry on oeis.org

0, 1, 8, 15, 64, 71, 120, 169, 512, 519, 568, 617, 960, 1009, 1352, 1695, 4096, 4103, 4152, 4201, 4544, 4593, 4936, 5279, 7680, 7729, 8072, 8415, 10816, 11159, 13560, 15961, 32768, 32775, 32824, 32873, 33216, 33265, 33608, 33951, 36352, 36401, 36744, 37087, 39488
Offset: 0

Views

Author

Omar E. Pol, Jun 14 2009

Keywords

Comments

First differences are in A161343. - Omar E. Pol, May 03 2015
From Gary W. Adamson, Aug 30 2016: (Start)
Let M =
1, 0, 0, 0, 0, ...
8, 0, 0, 0, 0, ...
7, 1, 0, 0, 0, ...
0, 8, 0, 0, 0, ...
0, 7, 1, 0, 0, ...
0, 0, 8, 0, 0, ...
0, 0, 7, 1, 0, ...
...
Then M^k converges to a single nonzero column giving the sequence.
The sequence with offset 1 divided by its aerated variant is (1, 8, 7, 0, 0, 0, ...). (End)

Links

Crossrefs

Cf. A160410, A160428, A161343, A006046, A130665, A116520, A130667, A116522, A116526, A116525, A360189.

Programs

  • Maple
    b:= proc(n) option remember; `if`(n<0, 0,
      b(n-1)+x^add(i, i=Bits[Split](n)))
    end:
a:= n-> subs(x=7, b(n-1)):
seq(a(n), n=0..44);  # Alois P. Heinz, Mar 06 2023
  • Mathematica
    A161342list[nmax_]:=Join[{0},Accumulate[7^DigitCount[Range[0,nmax-1],2,1]]];A161342list[100] (* Paolo Xausa, Aug 05 2023 *)

Formula

From Nathaniel Johnston, Nov 13 2010: (Start)
a(n) = Sum_{k=0..n-1} 7^A000120(k).
a(n) = 1 + 7 * Sum_{k=1..n-1} A151785(k), for n >= 1.
a(2^n) = 2^(3n).
(End)
a(n) = Sum_{k=0..floor(log_2(n))} 7^k*A360189(n-1,k). - Alois P. Heinz, Mar 06 2023

Extensions

More terms from Nathaniel Johnston, Nov 13 2010

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

A160119 A three-dimensional version of the cellular automaton A160118, using cubes.

Original entry on oeis.org

0, 1, 27, 35, 235, 243, 443, 499, 1899, 1907, 2107, 2163, 3563, 3619, 5019, 5411, 15211, 15219, 15419, 15475, 16875, 16931, 18331, 18723, 28523, 28579, 29979, 30371, 40171, 40563, 50363, 53107, 121707, 121715, 121915, 121971, 123371, 123427, 124827, 125219, 135019
Offset: 0

Views

Author

Omar E. Pol, May 05 2009

Keywords

Comments

Each cell has 26 neighbors.
Differs from A160379 in the same way that A160118 differs from A160117. - N. J. A. Sloane, Jan 01 2010

Links

Crossrefs

Cf. A000120, A139250, A139251, A151785, A160117, A160118, A160379.

Programs

  • Mathematica
    With[{d = 3}, 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) = 27 + 8*Sum_{k=1..n-1}A151785(k) + 200*Sum_{k=1..n-2}A151785(k), n >= 2.
a(2n) = 27 + 8*Sum_{k=1..n-1}A151785(k) + 200*Sum_{k=1..n-1}A151785(k), n >= 1.
In general, a d-dimensional version of the cellular automaton A160118 has its cell count given by the following formulas (where wt(k) = A000120(k)):
a(2n-1) = 3^d + (2^d)*Sum_{k=1..n-1}(2^d-1)^(wt(k)-1) + (2^d)*(3^d-2)*Sum_{k=1..n-2}(2^d-1)^(wt(k)-1), n >= 2.
a(2n) = 3^d + (2^d)*Sum_{k=1..n-1}(2^d-1)^(wt(k)-1) + (2^d)*(3^d-2)*Sum_{k=1..n-1}(2^d-1)^(wt(k)-1), n >= 1. (End)

Extensions

More terms from Omar E. Pol, May 11 2009
Edited by N. J. A. Sloane, Sep 05 2009
a(8)-a(32) from Nathaniel Johnston, Mar 24 2011
More terms from Amiram Eldar, Aug 01 2023

A161343 a(n) = 7^A000120(n).

Original entry on oeis.org

1, 7, 7, 49, 7, 49, 49, 343, 7, 49, 49, 343, 49, 343, 343, 2401, 7, 49, 49, 343, 49, 343, 343, 2401, 49, 343, 343, 2401, 343, 2401, 2401, 16807, 7, 49, 49, 343, 49, 343, 343, 2401, 49, 343, 343, 2401, 343, 2401, 2401, 16807, 49, 343, 343, 2401, 343, 2401, 2401, 16807, 343, 2401, 2401, 16807, 2401, 16807, 16807, 117649
Offset: 0

Views

Author

Omar E. Pol, Jun 14 2009

Keywords

Comments

Also first differences of A161342.
From Omar E. Pol, May 03 2015: (Start)
It appears that when A151785 is regarded as a triangle in which the row lengths are the powers of 2, this is what the rows converge to.
Also this is also a row of the square array A256140.
(End)

Examples

			From _Omar E. Pol_, May 03 2015: (Start)
Also, written as an irregular triangle in which the row lengths are the terms of A011782, the sequence begins:
1;
7;
7, 49;
7, 49, 49, 343;
7, 49, 49, 343, 49, 343, 343, 2401;
7, 49, 49, 343, 49, 343, 343, 2401, 49, 343, 343, 2401, 343, 2401, 2401, 16807;
...
Row sums give A055274.
Right border gives A000420.
(End)

Links

Crossrefs

Cf. A000420, A011782, A055274, A160410, A151785, A161411, A160428, A160429, A161342, A256140, A256141.
Rows of the array A256140 are: A000007, A000012, A001316, A048883, A102376, A256135, A256136, this sequence.

Programs

  • PARI
    a(n) = 7^hammingweight(n); \\ Omar E. Pol, May 03 2015

Formula

a(n) = A000420(A000120(n)). - Omar E. Pol, May 03 2015
G.f.: Product_{k>=0} (1 + 7*x^(2^k)). - Ilya Gutkovskiy, Mar 02 2017

Extensions

More terms from Sean A. Irvine, Mar 08 2011
New name from Omar E. Pol, May 03 2015
a(52)-a(63) from Omar E. Pol, May 16 2015

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
Showing 1-5 of 5 results.

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

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