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%I #37 Dec 13 2021 16:08:10
%S 1,3,5,15,29,55,93,247,477,887,1501,3959,7645,14199,24029,63351,
%T 122333,227191,384477,1013623,1957341,3635063,6151645,16217975,
%U 31317469,58161015,98426333,259487607,501079517,930576247,1574821341,4151801719,8017272285
%N Decimal representation of n-th iteration of the Rule 188 elementary cellular automaton starting with a single black cell.
%H Colin Barker, <a href="/A118173/b118173.txt">Table of n, a(n) for n = 0..1000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Rule188.html">Rule 188</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,0,16,0,-16).
%H <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%F a(n) = (1/30)*(-14 + 3*i*(2*i)^n + 55*2^n) for n odd,
%F a(n) = (1/15)*(-13 + 3*(2*i)^n + 25*2^n) for n even, where i = sqrt(-1).
%F From _Colin Barker_, Oct 08 2015: (Start)
%F G.f.: -(8*x^5-8*x^4-12*x^3-4*x^2-3*x-1) / ((x-1)*(x+1)*(2*x-1)*(2*x+1)*(4*x^2+1)).
%F a(n) = a(n-2) + 16*a(n-4) - 16*a(n-6) for n>5. (End)
%F E.g.f.: (1/15)*(6*sinh(x) + (5/2)*sinh(2x) + 25*exp(2x) - 13*exp(x)) + (1/10)*(2*cos(2x)-sin(2x)). - _G. C. Greubel_, Oct 08 2015
%F a(n) = floor(28*4^n/15) + 2^n - floor(28*2^n/15)*2^n. - _Karl V. Keller, Jr._, Nov 11 2021
%e 1; --> 1
%e 0, 1, 1; --> 3
%e 0, 0, 1, 0, 1; --> 5
%e 0, 0, 0, 1, 1, 1, 1; --> 15
%e 0, 0, 0, 0, 1, 1, 1, 0, 1; --> 29
%t clip[lst_] := Block[{p = Flatten@ Position[lst, 1]}, Take[lst, {Min@ p, Max@ p}]]; FromDigits[#, 2] & /@ Map[clip, CellularAutomaton[188, {{1}, 0}, 32]] (* _Michael De Vlieger_, Oct 08 2015 *)
%t RecurrenceTable[{a[n+6]==a[n+4] + 16*a[n+2] - 16*a[n], a[0]==1, a[1]==3, a[2]==5, a[3]==15, a[4]==29, a[5]==55}, a, {n,0,100}] (* _G. C. Greubel, Oct 08 2015 *)
%o (PARI) Vec(-(8*x^5-8*x^4-12*x^3-4*x^2-3*x-1)/((x-1)*(x+1)*(2*x-1)*(2*x+1)*(4*x^2+1)) + O(x^40)) \\ _Colin Barker_, Oct 08 2015
%o (Python) print([28*4**n//15 + 2**n - (28*2**n//15)*2**n for n in range(50)]) # _Karl V. Keller, Jr._, Nov 11 2021
%Y Cf. A118174, A265427.
%K nonn,base,easy
%O 0,2
%A _Eric W. Weisstein_, Apr 13 2006
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