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

0,2

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

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

Example

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

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 *)

Prog

(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
(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

Crossrefs

Cf. A118174, A265427.

Sequence in context: A340670 A284031 A284410 * A079450 A284482 A166956

Adjacent sequences: A118170 A118171 A118172 * A118174 A118175 A118176

Keyword

nonn,base,easy

Author

Eric W. Weisstein, Apr 13 2006

Status

approved