A070952 -id:A070952 - cp's OEIS frontend

A071032 Triangle read by rows giving successive states of cellular automaton generated by "Rule 86".

1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1

Offset: 0

Hans Havermann, May 26 2002

Row n has length 2n+1.

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; Chapter 3.

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened
Eric Weisstein's World of Mathematics, Rule 30
S. Wolfram, A New Kind of Science
Index to Elementary Cellular Automata
Index entries for sequences related to cellular automata

Cf. A005408 (row lengths), A070952 (row sums), A051023 (central terms), A070950 (mirror image, rule 30), A226464 (complemented, rule 149).

a071032 n k = a071032_tabf !! n !! k
a071032_row n = a071032_tabf !! n
a071032_tabf = map reverse a070950_tabf
-- Reinhard Zumkeller, Jun 08 2013

A071032list[rowmax_]:=MapIndexed[ArrayPad[#1,#2-rowmax-1]&,CellularAutomaton[86,{{1},0},rowmax]];A071032list[10] (* Generates 11 rows *) (* Paolo Xausa, Jun 01 2023 *)

T(n,k) = A070950(n,2*n+1-k), 0 <= k <= 2*n+1. - Reinhard Zumkeller, Jun 08 2013

Corrected by Hans Havermann, Jan 07 2012

A328105 Binary weight of A328104: a(n) = A000120(A110240(n) OR 2*A110240(n)).

Original entry on oeis.org

2, 4, 5, 8, 7, 12, 9, 15, 11, 17, 17, 20, 19, 26, 21, 29, 22, 27, 30, 33, 30, 34, 37, 40, 37, 39, 41, 49, 44, 49, 48, 53, 41, 56, 49, 64, 50, 62, 59, 66, 64, 60, 66, 69, 61, 77, 65, 73, 67, 74, 70, 89, 78, 87, 78, 94, 85, 88, 89, 100, 91, 101, 95, 110, 92, 85, 98, 102, 102, 102, 115, 109, 101, 105, 121, 118, 121, 129

Offset: 0

Antti Karttunen, Oct 05 2019

nonn

Cf. A000120, A110240, A163617, A269160, A328104.

Cf. also A070952, A328106, A328107, A328108, A328109.

A269160(n) = bitxor(n, bitor(2*n, 4*n));
A110240(n) = if(!n,1,A269160(A110240(n-1)));
A163617(n) = bitor(n,n<<1);
A328105(n) = hammingweight(A163617(A110240(n)));
\\ Use this one for writing b-files:
A328105write(up_to) = { my(s=1, n=0); for(n=0,up_to, write("b328105.txt", n, " ", hammingweight(bitor(s, s<<1))); s = A269160(s)); };

a(n) = A000120(A328104(n)) = A000120(A163617(A110240(n))).

For all n >= 0, A070952(a) < a(n) <= 2*A070952(n).

A328107 Binary weight of A327973.

Original entry on oeis.org

2, 4, 5, 6, 7, 8, 9, 13, 11, 13, 13, 14, 17, 18, 19, 23, 20, 23, 24, 27, 26, 24, 23, 30, 31, 29, 29, 31, 36, 35, 36, 37, 35, 34, 35, 42, 40, 46, 41, 50, 54, 48, 52, 47, 47, 53, 47, 51, 51, 54, 48, 51, 60, 55, 56, 64, 61, 60, 59, 68, 71, 67, 65, 78, 64, 63, 68, 72, 70, 74, 79, 89, 85, 77, 85, 76, 79, 83, 78, 90, 97, 82, 87, 81

Offset: 1

Antti Karttunen, Oct 05 2019

nonn

Cf. A000120, A110240, A269160, A327973.

Cf. also A070952, A328105, A328106, A328108, A328109.

A269160(n) = bitxor(n, bitor(2*n, 4*n)); \\ From A269160.
A110240(n) = if(!n,1,A269160(A110240(n-1)));
A327973(n) = bitxor(A110240(n), 2*A110240(n-1));
A328107(n) = hammingweight(bitxor(A110240(n), 2*A110240(n-1)));
\\ Use this one for writing b-files:
A328107write(up_to) = { my(s=1, t, n=0); for(n=1,up_to, t = A269160(s); write("b328107.txt", n, " ", hammingweight(bitxor(2*s, t))); s = t); };

a(n) = A000120(A327973(n)) = A000120(A110240(n) XOR 2*A110240(n-1)).

A328108 Binary weight of A327976.

Original entry on oeis.org

2, 4, 3, 6, 5, 12, 7, 11, 9, 13, 7, 12, 13, 20, 15, 23, 16, 19, 22, 25, 20, 28, 19, 30, 29, 39, 27, 29, 32, 37, 32, 37, 29, 38, 37, 38, 36, 44, 47, 44, 42, 46, 42, 53, 41, 49, 53, 47, 45, 58, 52, 55, 56, 65, 66, 60, 67, 56, 61, 64, 63, 77, 59, 66, 60, 67, 72, 72, 64, 84, 57, 81, 63, 79, 67, 92, 77, 77, 74, 80, 81, 88, 77

Offset: 1

Antti Karttunen, Oct 05 2019

nonn

Cf. A000120, A030101, A110240, A265281, A269160, A327976.

Cf. also A070952, A328105, A328106, A328107, A328109.

A269160(n) = bitxor(n, bitor(2*n, 4*n)); \\ From A269160.
A110240(n) = if(!n,1,A269160(A110240(n-1)));
A030101(n) = if(n<1,0,subst(Polrev(binary(n)),x,2));
A328108(n) = hammingweight(bitxor(A110240(n), 2*A030101(A110240(n-1))));
\\ Use this one for writing b-files:
A328108write(up_to) = { my(s=1, t, n=0); for(n=1,up_to, t = A269160(s); write("b328108.txt", n, " ", hammingweight(bitxor(2*A030101(s), t))); s = t); };

a(n) = A000120(A327976(n)).
a(n) = A000120(A110240(n) XOR 2*A265281(n-1)).
a(n) = A000120(A110240(n) XOR 2*A030101(A110240(n-1))).

A110267 Total number of black cells at the first n generations of a single black cell following Wolfram's Rule 30 cellular automaton.

Original entry on oeis.org

1, 4, 7, 13, 17, 26, 31, 43, 50, 62, 73, 87, 99, 118, 131, 153, 168, 187, 207, 231, 252, 275, 298, 326, 352, 379, 405, 438, 468, 502, 533, 572, 598, 637, 666, 712, 744, 788, 826, 871, 918, 959, 1004, 1053, 1091, 1146, 1188, 1239, 1283, 1336, 1379, 1438, 1490

Offset: 0

Alexandre Wajnberg, Sep 06 2005

At each generation, "looking back", one can see "behind", groups of black cells: total number of black cells (cumulative sum of first n terms of A070952).

			a(1)=1 because one black cell;
a(2)=4 because there are now 3 contiguous black cell connected to the first one, which form one only black surface of 4 cells;
a(3)=7 because appear three black cells: 4+3=7
From _Michael De Vlieger_, Dec 16 2015: (Start)
First 12 rows, replacing "0" with "." for better visibility of ON cells, followed by the total number of ON cells per row, and the running total up to that row:
                1                  =  1 ->   1
              1 1 1                =  3 ->   4
            1 1 . . 1              =  3 ->   7
          1 1 . 1 1 1 1            =  6 ->  13
        1 1 . . 1 . . . 1          =  4 ->  17
      1 1 . 1 1 1 1 . 1 1 1        =  9 ->  26
    1 1 . . 1 . . . . 1 . . 1      =  5 ->  31
  1 1 . 1 1 1 1 . . 1 1 1 1 1 1    = 12 ->  43
1 1 . . 1 . . . 1 1 1 . . . . . 1  =  7 ->  50
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Rule 30.
Index entries for sequences related to cellular automata

Cf. A070950, A051023, A092539, A092540, A070952, A100053, A100054, A100055, A094603, A094604, A000225, A074890.

See A265704 for an essentially identical sequence.

a110267 n = a110267_list !! (n-1)
a110267_list = scanl1 (+) a070952_list
-- Reinhard Zumkeller, Jun 08 2013

Accumulate[Total /@ CellularAutomaton[30, {{1}, 0}, 52]] (* Michael De Vlieger, Dec 16 2015 *)

Offset changed by Reinhard Zumkeller, Jun 08 2013

A246025 Record high water marks in A246024.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 11, 15, 16, 22, 24, 27, 33, 35, 42, 51, 53, 54, 58, 64, 78, 104, 132, 133, 139, 151, 164, 168, 204, 223, 242, 249, 256, 261, 289, 349, 385, 404, 438, 456, 460, 565, 579, 641, 661, 747, 800, 921, 929, 1012, 1053, 1112, 1249, 1251, 1384

Offset: 1

N. J. A. Sloane, Aug 14 2014

nonn

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..101

Cf. A070952, A246023, A246024, A246026.

a(32)-a(57) from Hiroaki Yamanouchi, Sep 12 2014

A246026 Positions of record high water marks in A246024.

Original entry on oeis.org

0, 1, 3, 5, 7, 13, 15, 31, 35, 63, 85, 91, 171, 179, 267, 315, 355, 526, 614, 699, 819, 1355, 1443, 1827, 1987, 5306, 5506, 5690, 5882, 6112, 7295, 10266, 12506, 13303, 13518, 17007, 18382, 20430, 23166, 24898, 34322, 36126, 42526, 43390, 43630, 48046, 75118

Offset: 1

N. J. A. Sloane, Aug 14 2014

nonn

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..101

Cf. A070952, A246023, A246024, A246025.

a(32)-a(47) from Hiroaki Yamanouchi, Sep 12 2014

A246027 a(n) = n - A071049(n).

Original entry on oeis.org

-1, -1, -1, 0, -1, 2, 1, 1, 0, 4, 4, 3, 4, 5, 3, 4, 3, 8, 7, 8, 7, 7, 6, 9, 10, 12, 13, 10, 6, 9, 14, 14, 8, 14, 20, 16, 11, 19, 18, 14, 16, 22, 18, 12, 17, 19, 22, 25, 16, 18, 22, 27, 23, 19, 24, 24, 19, 23, 24, 23, 25, 27, 27, 27, 21, 25, 30, 29, 31, 30, 30, 27, 28, 31, 29, 27, 33, 30, 42, 42, 34

Offset: 0

N. J. A. Sloane, Aug 14 2014

sign

Note that this is much larger than the analogous sequence for Rule 30 (see A070952, A246024). This is because it appears that A071049 only grows as c*n, where c is about 3/5, whereas A070952 is roughly equal to n.

N. J. A. Sloane, Table of n, a(n) for n = 0..10000

Cf. A071049, A246024, A070952.

A265703 Number of OFF (white) cells in the n-th iteration of the "Rule 135" elementary cellular automaton starting with a single ON (black) cell.

Original entry on oeis.org

0, 1, 3, 3, 6, 4, 9, 5, 12, 7, 12, 11, 14, 12, 19, 13, 22, 15, 19, 20, 24, 21, 23, 23, 28, 26, 27, 26, 33, 30, 34, 31, 39, 26, 39, 29, 46, 32, 44, 38, 45, 47, 41, 45, 49, 38, 55, 42, 51, 44, 53, 43, 59, 52, 60, 49, 65, 57, 60, 56, 69, 61, 70, 59, 78, 64, 56

Offset: 0

Robert Price, Dec 13 2015

This appears to be (apart from the initial zero and an index shift) the same as A070952. - R. J. Mathar, Dec 16 2015

Presumably that will not be difficult to prove. - N. J. A. Sloane, Jan 09 2016

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

Robert Price, Table of n, a(n) for n = 0..999
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
Index entries for sequences related to cellular automata
Index to Elementary Cellular Automata

Cf. A070952, A265695.

A110266 Number of blocks of ON cells in n-th row of triangle generated by Wolfram's "Rule 30".

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 4, 3, 4, 5, 6, 6, 7, 7, 8, 7, 7, 8, 10, 9, 9, 11, 14, 12, 11, 12, 15, 16, 14, 15, 17, 14, 15, 17, 20, 18, 18, 18, 21, 21, 17, 19, 21, 20, 23, 22, 23, 22, 23, 21, 27, 30, 26, 27, 29, 29, 28, 28, 33, 31, 30, 31, 36, 32, 28, 29, 33, 33, 33, 35

Offset: 1

Alexandre Wajnberg, Sep 06 2005

Old name was "Number of trees appearing at n-th generation of a black cell following Wolfram's Rule 30 cellular automaton."

At each generation, "looking back", one can see "behind", groups (sort of black isles) of contiguous black cells which after a while appear to be trees growing. It should be possible to describe each one of them in terms of trees theory.

			a(1)=1 because one black cell;
a(2)=1 because there are now 3 contiguous black cell connected to the first one, which forms one only black surface;
a(3)=2 because two black cells are now connected to the preceding black surface and another black cell appears, which is isolated, so we have two separate black surfaces: 2.
From _Charlie Neder_, Feb 06 2019: (Start)
Rule 30 triangle begins:
                     1
                    111
                   11  1
                  11 1111
                 11  1   1
                11 1111 111
               11  1    1  1
              11 1111  111111
             11  1   111     1
and the number of blocks of ON cells in each row is 1, 1, 2, 2, 3, 3, 4, 3, 4, ... (End)

Charlie Neder, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Rule 30.

Cf. A070950, A051023, A092539, A092540, A070952, A100053, A100054, A100055, A094603, A094604, A000225, A074890.

New name and a(17)-a(70) from Charlie Neder, Feb 06 2019

cp's OEIS Frontend

A071032 Triangle read by rows giving successive states of cellular automaton generated by "Rule 86".

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A328105 Binary weight of A328104: a(n) = A000120(A110240(n) OR 2*A110240(n)).

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A328107 Binary weight of A327973.

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A328108 Binary weight of A327976.

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A110267 Total number of black cells at the first n generations of a single black cell following Wolfram's Rule 30 cellular automaton.

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A246025 Record high water marks in A246024.

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A246026 Positions of record high water marks in A246024.

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A246027 a(n) = n - A071049(n).

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A265703 Number of OFF (white) cells in the n-th iteration of the "Rule 135" elementary cellular automaton starting with a single ON (black) cell.

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