A118173 Decimal representation of n-th iteration of the Rule 188 elementary cellular automaton starting with a single black cell.
1, 3, 5, 15, 29, 55, 93, 247, 477, 887, 1501, 3959, 7645, 14199, 24029, 63351, 122333, 227191, 384477, 1013623, 1957341, 3635063, 6151645, 16217975, 31317469, 58161015, 98426333, 259487607, 501079517, 930576247, 1574821341, 4151801719, 8017272285
Offset: 0
Examples
1; --> 1 0, 1, 1; --> 3 0, 0, 1, 0, 1; --> 5 0, 0, 0, 1, 1, 1, 1; --> 15 0, 0, 0, 0, 1, 1, 1, 0, 1; --> 29
Links
- Colin Barker, Table of n, a(n) for n = 0..1000
- Eric Weisstein's World of Mathematics, Rule 188
- Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
- Index entries for linear recurrences with constant coefficients, signature (0,1,0,16,0,-16).
- Index entries for sequences related to cellular automata
Programs
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Mathematica
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 *) 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 *) -
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
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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
Formula
a(n) = (1/30)*(-14 + 3*i*(2*i)^n + 55*2^n) for n odd,
a(n) = (1/15)*(-13 + 3*(2*i)^n + 25*2^n) for n even, where i = sqrt(-1).
From Colin Barker, Oct 08 2015: (Start)
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)).
a(n) = a(n-2) + 16*a(n-4) - 16*a(n-6) for n>5. (End)
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
a(n) = floor(28*4^n/15) + 2^n - floor(28*2^n/15)*2^n. - Karl V. Keller, Jr., Nov 11 2021