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

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

0,2

Links

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).

Formula

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

Example

            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)

Mathematica

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 *)

Prog

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

Crossrefs

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

Sequence in context: A124828 A296646 A037094 * A072261 A066744 A037576

Adjacent sequences: A118168 A118169 A118170 * A118172 A118173 A118174

Keyword

nonn,base,easy

Author

Eric W. Weisstein, Apr 13 2006

Status

approved