A071037 -id:A071037 - cp's OEIS frontend

A118172 Duplicate of A071037.

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

Offset: 0

Eric W. Weisstein, Apr 13 2006

dead

A029578 The natural numbers interleaved with the even numbers.

Original entry on oeis.org

0, 0, 1, 2, 2, 4, 3, 6, 4, 8, 5, 10, 6, 12, 7, 14, 8, 16, 9, 18, 10, 20, 11, 22, 12, 24, 13, 26, 14, 28, 15, 30, 16, 32, 17, 34, 18, 36, 19, 38, 20, 40, 21, 42, 22, 44, 23, 46, 24, 48, 25, 50, 26, 52, 27, 54, 28, 56, 29, 58, 30, 60, 31, 62, 32, 64, 33, 66, 34, 68, 35, 70, 36, 72

Offset: 0

N. J. A. Sloane

a(n) = number of ordered, length two, compositions of n with at least one odd summand - Len Smiley, Nov 25 2001
Also number of 0's in n-th row of triangle in A071037. - Hans Havermann, May 26 2002
For n > 2: a(n) = number of odd terms in row n-2 of triangle A265705. - Reinhard Zumkeller, Dec 15 2015

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for two-way infinite sequences
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A065423 (at least one even summand).

Cf. A001477, A005843, A009531, A071037, A211538 (partial sums), A265705.

import Data.List (transpose)
a029578 n =  (n - n `mod` 2) `div` (2 - n `mod` 2)
a029578_list = concat $ transpose [a001477_list, a005843_list]
-- Reinhard Zumkeller, Nov 27 2012

A029578:= func< n | (n + (n-2)*(n mod 2))/2 >;
[A029578(n): n in [0..80]]; // G. C. Greubel, Jan 22 2025

With[{nn=40},Riffle[Range[0,nn],Range[0,2nn,2]]] (* or *) LinearRecurrence[ {0,2,0,-1},{0,0,1,2},80] (* Harvey P. Dale, Aug 23 2015 *)

PARI
```
a(n)=if(n%2,n-1,n/2)
```

def A029578(n): return (n + (n-2)*(n%2))//2
print([A029578(n) for n in range(81)]) # G. C. Greubel, Jan 22 2025

a(n) = (3*n - 2 - (-1)^n*(n - 2))/4.
a(n+4) = 2*a(n+2) - a(n).
G.f.: x^2*(1 + 2*x)/(1-x^2)^2.
a(n) = floor((n+1)/2) + (n is odd)*floor((n+1)/2).
a(n) = (n - n mod 2)/(2 - n mod 2). - Reinhard Zumkeller, Jul 30 2002
a(n) = floor(n/2)*binomial(2, mod(n, 2)) - Paul Barry, May 25 2003
a(2*n) = n.
a(2*n-1) = 2*n-2.
a(-n) = -A065423(n+2).
a(n) = Sum_{k=0..floor((n-2)/2)} (C(n-k-2, k) mod 2)((1+(-1)^k)/2)*2^A000120(n-2k-2). - Paul Barry, Jan 06 2005
a(n) = Sum_{k=0..n-2} gcd(n-k-1, k+1). - Paul Barry, May 03 2005
For n>6: a(n) = floor(a(n-1)*a(n-2)/a(n-3)). - Reinhard Zumkeller, Mar 06 2011
E.g.f.: (1/4)*((x+2)*exp(-x) + (3*x-2)*exp(x)). - G. C. Greubel, Jan 22 2025

Explicated definition by Reinhard Zumkeller, Nov 27 2012

Title simplified by Sean A. Irvine, Feb 29 2020

A118171 Decimal representation of n-th iteration of the Rule 158 elementary cellular automaton starting with a single black cell.

Original entry on oeis.org

1, 7, 29, 115, 477, 1843, 7645, 29491, 122333, 471859, 1957341, 7549747, 31317469, 120795955, 501079517, 1932735283, 8017272285, 30923764531, 128276356573, 494780232499, 2052421705181, 7916483719987, 32838747282909, 126663739519795, 525419956526557

Offset: 0

Eric W. Weisstein, Apr 13 2006

			            1;
         1, 1, 1;
      1, 1, 1, 0, 1;
   1, 1, 1, 0, 0, 1, 1;
1, 1, 1, 0, 1, 1, 1, 0, 1;
...
From _Michael De Vlieger_, Oct 08 2015: (Start)
First 8 rows, representing ON cells as "1", OFF cells within the bounds
of ON cells as "0", interpreted as a binary number at left, the decimal
equivalent appearing at right:
                   1 =      1
                 111 =      7
              1 1101 =     29
            111 0011 =    115
         1 1101 1101 =    477
       111 0011 0011 =   1843
    1 1101 1101 1101 =   7645
  111 0011 0011 0011 =  29491
11101 1101 1101 1101 = 122333
(End)

Colin Barker, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Rule 158
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
Index entries for sequences related to cellular automata
Index entries for linear recurrences with constant coefficients, signature (0,17,0,-16).

Cf. A071037 (cells), A265379 (binary).

Table[(-16 + (-4)^n - 10 (-1)^n + 55*4^n)/30, {n, 0, 24}] (* or *)
clip[lst_] := Block[{p = Flatten@ Position[lst, 1]}, Take[lst, {Min@ p, Max@ p}]]; FromDigits[#, 2] & /@ Map[clip, CellularAutomaton[158, {{1}, 0}, 24]] (* Michael De Vlieger, Oct 08 2015 *)

Vec(-(4*x^3-12*x^2-7*x-1)/((x-1)*(x+1)*(4*x-1)*(4*x+1)) + O(x^30)) \\ Colin Barker, Oct 08 2015

vector(100, n, n--; (1/30)*(-16+(-4)^n-10*(-1)^n+55*4^n)) \\ Altug Alkan, Oct 08 2015

print([27*4**n//15 if n%2 else 28*4**n//15 for n in range(50)]) # Karl V. Keller, Jr., May 07 2022

a(n) = (1/30)*(-16+(-4)^n-10*(-1)^n+55*4^n).
From Colin Barker, Oct 08 2015: (Start)
a(n) = 17*a(n-2) - 16*a(n-4) for n>3.
G.f.: -(4*x^3-12*x^2-7*x-1) / ((x-1)*(x+1)*(4*x-1)*(4*x+1)).
(End)
a(n) = floor(28*4^n/15) for even n>=0; a(n) = floor(27*4^n/15) for odd n. - Karl V. Keller, Jr., Oct 09 2020

A265381 Decimal representation of the middle column of the "Rule 158" elementary cellular automaton starting with a single ON (black) cell.

Original entry on oeis.org

1, 3, 7, 14, 29, 59, 119, 238, 477, 955, 1911, 3822, 7645, 15291, 30583, 61166, 122333, 244667, 489335, 978670, 1957341, 3914683, 7829367, 15658734, 31317469, 62634939, 125269879, 250539758, 501079517, 1002159035, 2004318071, 4008636142, 8017272285

Offset: 0

Robert Price, Dec 07 2015

			From _Michael De Vlieger_, Dec 09 2015: (Start)
First 8 rows at left, ignoring "0" outside of range of 1's, the center column values in parentheses. The center column values up to that row are concatenated then converted into decimal at right:
             Rule 158                   Binary     Decimal
                (1)                 ->         1 =   1
              1 (1) 1               ->        11 =   3
            1 1 (1) 0 1             ->       111 =   7
          1 1 1 (0) 0 1 1           ->      1110 =  14
        1 1 1 0 (1) 1 1 0 1         ->     11101 =  29
      1 1 1 0 0 (1) 1 0 0 1 1       ->    111011 =  59
    1 1 1 0 1 1 (1) 0 1 1 1 0 1     ->   1110111 = 119
  1 1 1 0 0 1 1 (0) 0 1 1 0 0 1 1   ->  11101110 = 238
1 1 1 0 1 1 1 0 (1) 1 1 0 1 1 1 0 1 -> 111011101 = 477
(End)

Robert Price, Table of n, a(n) for n = 0..999
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
Eric Weisstein's World of Mathematics, Rule 158
Stephen Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.
Index entries for sequences related to cellular automata
Index to Elementary Cellular Automata
Index entries for linear recurrences with constant coefficients, signature (2,0,0,1,-2).

Cf. A071037, A265380 (binary).

f[n_] := Block[{w = {}}, Do[AppendTo[w, Boole[Mod[k, 4] != 3]], {k, 0, n}]; FromDigits[w, 2]]; Table[f@ n, {n, 0, 32}] (* Michael De Vlieger, Dec 09 2015 *)

print([7*2**(n+2)//15 for n in range(34)]) # Karl V. Keller, Jr., Oct 01 2020

From Colin Barker, Dec 07 2015 and Apr 16 2019: (Start)
a(n) = (-45+5*(-1)^n-(6-i*3)*(-i)^n-(6+3*i)*i^n+7*2^(4+n))/60 where i = sqrt(-1).
a(n) = 2*a(n-1)+a(n-4)-2*a(n-5) for n>4.
G.f.: (1+x+x^2) / ((1-x)*(1+x)*(1-2*x)*(1+x^2)).
(End)
a(n) = floor(7*2^(n+2)/15) for n>=0. - Karl V. Keller, Jr., Oct 01 2020

A265379 Binary representation of the n-th iteration of the "Rule 158" elementary cellular automaton starting with a single ON (black) cell.

Original entry on oeis.org

1, 111, 11101, 1110011, 111011101, 11100110011, 1110111011101, 111001100110011, 11101110111011101, 1110011001100110011, 111011101110111011101, 11100110011001100110011, 1110111011101110111011101, 111001100110011001100110011, 11101110111011101110111011101

Offset: 0

Robert Price, Dec 07 2015

			From _Michael De Vlieger_, Dec 09 2015: (Start)
First 12 rows:
                        1
                      1 1 1
                    1 1 1 0 1
                  1 1 1 0 0 1 1
                1 1 1 0 1 1 1 0 1
              1 1 1 0 0 1 1 0 0 1 1
            1 1 1 0 1 1 1 0 1 1 1 0 1
          1 1 1 0 0 1 1 0 0 1 1 0 0 1 1
        1 1 1 0 1 1 1 0 1 1 1 0 1 1 1 0 1
      1 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1
    1 1 1 0 1 1 1 0 1 1 1 0 1 1 1 0 1 1 1 0 1
  1 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1
1 1 1 0 1 1 1 0 1 1 1 0 1 1 1 0 1 1 1 0 1 1 1 0 1
(End)

Robert Price, Table of n, a(n) for n = 0..499
Eric Weisstein's World of Mathematics, Rule 158
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
Stephen Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.
Index entries for sequences related to cellular automata
Index to Elementary Cellular Automata
Index entries for linear recurrences with constant coefficients, signature (0,10001,0,-10000).

Cf. A071037 (cells), A118171 (decimal).

rule = 158; rows = 20; Table[FromDigits[Table[Take[CellularAutomaton[rule,{{1},0}, rows-1, {All,All}][[k]], {rows-k+1, rows+k-1}], {k,1,rows}][[k]]], {k,1,rows}]

print([(11100 - (n%2))*100**n//9999 for n in range(30)]) # Karl V. Keller, Jr., Sep 20 2021

From Colin Barker, Dec 14 2015 and Apr 18 2019: (Start)
a(n) = 10001*a(n-2) - 10000*a(n-4) for n>3.
G.f.: (1+111*x+1100*x^2-100*x^3) / ((1-x)*(1+x)*(1-100*x)*(1+100*x)).
(End)
a(n) = floor((11100 - (n mod 2))*100^n/9999). - Karl V. Keller, Jr., Sep 20 2021

A265380 Binary representation of the middle column of the "Rule 158" elementary cellular automaton starting with a single ON (black) cell.

Original entry on oeis.org

1, 11, 111, 1110, 11101, 111011, 1110111, 11101110, 111011101, 1110111011, 11101110111, 111011101110, 1110111011101, 11101110111011, 111011101110111, 1110111011101110, 11101110111011101, 111011101110111011, 1110111011101110111, 11101110111011101110

Offset: 0

Robert Price, Dec 07 2015

			From _Michael De Vlieger_, Dec 09 2015: (Start)
First 8 rows at left, ignoring "0" outside of range of 1's, the center column values in parentheses, and at right the value of center column cells up to that row :
                        (1)                          -> 1
                      1 (1) 1                        -> 11
                    1 1 (1) 0 1                      -> 111
                  1 1 1 (0) 0 1 1                    -> 1110
                1 1 1 0 (1) 1 1 0 1                  -> 11101
              1 1 1 0 0 (1) 1 0 0 1 1                -> 111011
            1 1 1 0 1 1 (1) 0 1 1 1 0 1              -> 1110111
          1 1 1 0 0 1 1 (0) 0 1 1 0 0 1 1            -> 11101110
        1 1 1 0 1 1 1 0 (1) 1 1 0 1 1 1 0 1          -> 111011101
      1 1 1 0 0 1 1 0 0 (1) 1 0 0 1 1 0 0 1 1        -> 1110111011
    1 1 1 0 1 1 1 0 1 1 (1) 0 1 1 1 0 1 1 1 0 1      -> 11101110111
  1 1 1 0 0 1 1 0 0 1 1 (0) 0 1 1 0 0 1 1 0 0 1 1    -> 111011101110
1 1 1 0 1 1 1 0 1 1 1 0 (1) 1 1 0 1 1 1 0 1 1 1 0 1  -> 1110111011101
(End)

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. A071037, A265381.

f[n_] := Block[{w = {}}, Do[AppendTo[w, Boole[Mod[k, 4] != 3]], {k, 0, n}]; FromDigits@ w]; Table[f@ n, {n, 0, 19}] (* Michael De Vlieger, Dec 09 2015 *)

Conjectures from Colin Barker, Dec 14 2015 and Apr 18 2019: (Start)
a(n) = 10*a(n-1) + a(n-4) - 10*a(n-5) for n>4.
G.f.: (1+x+x^2) / ((1-x)*(1+x)*(1-10*x)*(1+x^2)).
(End)

Removed an unjustified claim that Colin Barker's conjectures are correct. Removed 2 programs based on conjectures. - N. J. A. Sloane, Jun 13 2022

A071054 a(2n)=3n+1, a(2n+1)=2n+2.

Original entry on oeis.org

1, 3, 4, 5, 7, 7, 10, 9, 13, 11, 16, 13, 19, 15, 22, 17, 25, 19, 28, 21, 31, 23, 34, 25, 37, 27, 40, 29, 43, 31, 46, 33, 49, 35, 52, 37, 55, 39, 58, 41, 61, 43, 64, 45, 67, 47, 70, 49, 73, 51, 76, 53, 79, 55, 82, 57, 85, 59, 88, 61, 91, 63, 94, 65, 97, 67, 100, 69

Offset: 0

Hans Havermann, May 26 2002

nonn

Number of ON cells at n-th generation of 1-D CA defined by Rule 158, starting with a single ON cell at generation 0. Equivalently, number of 1's in n-th row of triangle in A071037. - N. J. A. Sloane, Aug 10 2014

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

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
A. J. Macfarlane, Generating functions for integer sequences defined by the evolution of cellular automata with even rule numbers, 2016.
N. J. A. Sloane, Illustration of first 20 generations of Rule 158
Index entries for sequences related to cellular automata

Cf. A029578.

CoefficientList[Series[(-x^3 + 2 x^2 + 3 x + 1)/(1 - x^2)^2, {x, 0, 100}], x] (* Vincenzo Librandi, Aug 11 2014 *)
ArrayPlot[CellularAutomaton[158, {{1}, 0}, 20]] (* N. J. A. Sloane, Aug 11 2014 *)

G.f.: (-x^3+2x^2+3x+1)/(1-x^2)^2.

a(n) = (5/4)*n + 3/2 + (n/4 - 1/2)*(-1)^n. - Robert Israel, Aug 11 2014

Simpler definition from N. J. A. Sloane, Aug 11 2014

A265382 Total number of ON (black) cells after n iterations of the "Rule 158" elementary cellular automaton starting with a single ON (black) cell.

Original entry on oeis.org

1, 4, 8, 13, 20, 27, 37, 46, 59, 70, 86, 99, 118, 133, 155, 172, 197, 216, 244, 265, 296, 319, 353, 378, 415, 442, 482, 511, 554, 585, 631, 664, 713, 748, 800, 837, 892, 931, 989, 1030, 1091, 1134, 1198, 1243, 1310, 1357, 1427, 1476, 1549, 1600, 1676, 1729

Offset: 0

Robert Price, Dec 07 2015

			From _Michael De Vlieger_, Dec 09 2015: (Start)
First 12 rows, replacing "0" with "." for better visibility of ON cells, followed by the total number of 1's per row, and the running total up to that row:
                        1                          =  1 ->   1
                      1 1 1                        =  3 ->   4
                    1 1 1 . 1                      =  4 ->   8
                  1 1 1 . . 1 1                    =  5 ->  13
                1 1 1 . 1 1 1 . 1                  =  7 ->  20
              1 1 1 . . 1 1 . . 1 1                =  7 ->  27
            1 1 1 . 1 1 1 . 1 1 1 . 1              = 10 ->  37
          1 1 1 . . 1 1 . . 1 1 . . 1 1            =  9 ->  46
        1 1 1 . 1 1 1 . 1 1 1 . 1 1 1 . 1          = 13 ->  59
      1 1 1 . . 1 1 . . 1 1 . . 1 1 . . 1 1        = 11 ->  70
    1 1 1 . 1 1 1 . 1 1 1 . 1 1 1 . 1 1 1 . 1      = 16 ->  86
  1 1 1 . . 1 1 . . 1 1 . . 1 1 . . 1 1 . . 1 1    = 13 ->  99
1 1 1 . 1 1 1 . 1 1 1 . 1 1 1 . 1 1 1 . 1 1 1 . 1  = 19 -> 118
(End)

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. A071037.

rule = 158; rows = 30; Table[Total[Take[Table[Total[Table[Take[CellularAutomaton[rule,{{1},0},rows-1,{All,All}][[k]],{rows-k+1,rows+k-1}],{k,1,rows}][[k]]],{k,1,rows}],k]],{k,1,rows}]
Accumulate[Count[#, n_ /; n == 1] & /@ CellularAutomaton[158, {{1}, 0}, 51]] (* Michael De Vlieger, Dec 09 2015 *)

Conjectures from Colin Barker, Dec 07 2015 and Apr 18 2019: (Start)
a(n) = 1/16*(10*n^2+2*(-1)^n*n+34*n-3*(-1)^n+19).
a(n) = 1/16*(10*n^2+36*n+16) for n even.
a(n) = 1/16*(10*n^2+32*n+22) for n odd.
a(n) = 2*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4) for n>4.
G.f.: (1+3*x+2*x^2-x^3) / ((1-x)^3*(1+x)^2).
(End)

cp's OEIS Frontend

A118172 Duplicate of A071037.

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A029578 The natural numbers interleaved with the even numbers.

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A118171 Decimal representation of n-th iteration of the Rule 158 elementary cellular automaton starting with a single black cell.

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A265381 Decimal representation of the middle column of the "Rule 158" elementary cellular automaton starting with a single ON (black) cell.

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A265379 Binary representation of the n-th iteration of the "Rule 158" elementary cellular automaton starting with a single ON (black) cell.

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A265380 Binary representation of the middle column of the "Rule 158" elementary cellular automaton starting with a single ON (black) cell.

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A071054 a(2n)=3n+1, a(2n+1)=2n+2.

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A265382 Total number of ON (black) cells after n iterations of the "Rule 158" elementary cellular automaton starting with a single ON (black) cell.