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
Examples
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)
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).
Programs
-
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 *) -
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
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