A160428 -id:A160428 - cp's OEIS frontend

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

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

Omar E. Pol, Jun 14 2009

nonn

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)

Alois P. Heinz, Table of n, a(n) for n = 0..16383
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata

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

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

A161342list[nmax_]:=Join[{0},Accumulate[7^DigitCount[Range[0,nmax-1],2,1]]];A161342list[100] (* Paolo Xausa, Aug 05 2023 *)

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

More terms from Nathaniel Johnston, Nov 13 2010

A160429 First differences of A160428.

Original entry on oeis.org

8, 56, 56, 392, 56, 392, 392, 2744, 56, 392, 392, 2744, 392, 2744, 2744, 19208, 56, 392, 392, 2744, 392, 2744, 2744, 19208, 392, 2744, 2744, 19208, 2744, 19208, 19208, 134456, 56, 392, 392, 2744, 392, 2744, 2744, 19208, 392, 2744, 2744, 19208, 2744, 19208, 19208

Offset: 1

Omar E. Pol, Jun 01 2009

nonn

Cf. A160411, A160428.

a(n) = 8*7^hammingweight(n-1); \\ Jinyuan Wang, Mar 14 2020

a(n) = 8 * 7^A000120(n-1) for n>=1.

Formula and more terms from Nathaniel Johnston, Nov 14 2010

More terms from Jinyuan Wang, Mar 14 2020

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

Omar E. Pol, Jun 14 2009

nonn

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

Jinyuan Wang, Table of n, a(n) for n = 0..1000
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata

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

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

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

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

Omar E. Pol, Jun 14 2009

nonn

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)

			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)

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.

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

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

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

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

Omar E. Pol, Sep 20 2009

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for sequences related to cellular automata
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

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

LinearRecurrence[{0,3,0,-3,0,1},{1,26,8,200,26,620,56,1304},50] (* Harvey P. Dale, May 19 2025 *)

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

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)

Formula and more terms from Nathaniel Johnston, Nov 15 2010

More terms from Colin Barker, Sep 17 2013

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

Omar E. Pol, Sep 20 2009

Index entries for sequences related to cellular automata

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

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

a(8)-a(32) from Nathaniel Johnston, Mar 24 2011

cp's OEIS Frontend

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

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A160429 First differences of A160428.

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

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

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A163989 First differences of A160379.

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A163987 First differences of A160119.

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