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

%I A118171 #43 Feb 16 2025 08:33:00
%S A118171 1,7,29,115,477,1843,7645,29491,122333,471859,1957341,7549747,
%T A118171 31317469,120795955,501079517,1932735283,8017272285,30923764531,
%U A118171 128276356573,494780232499,2052421705181,7916483719987,32838747282909,126663739519795,525419956526557
%N A118171 Decimal representation of n-th iteration of the Rule 158 elementary cellular automaton starting with a single black cell.
%H A118171 Colin Barker, <a href="/A118171/b118171.txt">Table of n, a(n) for n = 0..1000</a>
%H A118171 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Rule158.html">Rule 158</a>
%H A118171 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A118171 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A118171 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,17,0,-16).
%F A118171 a(n) = (1/30)*(-16+(-4)^n-10*(-1)^n+55*4^n).
%F A118171 From _Colin Barker_, Oct 08 2015: (Start)
%F A118171 a(n) = 17*a(n-2) - 16*a(n-4) for n>3.
%F A118171 G.f.: -(4*x^3-12*x^2-7*x-1) / ((x-1)*(x+1)*(4*x-1)*(4*x+1)).
%F A118171 (End)
%F A118171 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
%e A118171             1;
%e A118171          1, 1, 1;
%e A118171       1, 1, 1, 0, 1;
%e A118171    1, 1, 1, 0, 0, 1, 1;
%e A118171 1, 1, 1, 0, 1, 1, 1, 0, 1;
%e A118171 ...
%e A118171 From _Michael De Vlieger_, Oct 08 2015: (Start)
%e A118171 First 8 rows, representing ON cells as "1", OFF cells within the bounds
%e A118171 of ON cells as "0", interpreted as a binary number at left, the decimal
%e A118171 equivalent appearing at right:
%e A118171                    1 =      1
%e A118171                  111 =      7
%e A118171               1 1101 =     29
%e A118171             111 0011 =    115
%e A118171          1 1101 1101 =    477
%e A118171        111 0011 0011 =   1843
%e A118171     1 1101 1101 1101 =   7645
%e A118171   111 0011 0011 0011 =  29491
%e A118171 11101 1101 1101 1101 = 122333
%e A118171 (End)
%t A118171 Table[(-16 + (-4)^n - 10 (-1)^n + 55*4^n)/30, {n, 0, 24}] (* or *)
%t A118171 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 *)
%o A118171 (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
%o A118171 (PARI) vector(100, n, n--; (1/30)*(-16+(-4)^n-10*(-1)^n+55*4^n)) \\ _Altug Alkan_, Oct 08 2015
%o A118171 (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
%Y A118171 Cf. A071037 (cells), A265379 (binary).
%K A118171 nonn,base,easy
%O A118171 0,2
%A A118171 _Eric W. Weisstein_, Apr 13 2006