A266284 Decimal representation of the n-th iteration of the "Rule 13" elementary cellular automaton starting with a single ON (black) cell.

1, 2, 5, 106, 21, 1962, 85, 32426, 341, 522922, 1365, 8383146, 5461, 134195882, 21845, 2147396266, 87381, 34359388842, 349525, 549754415786, 1398101, 8796087429802, 5592405, 140737465985706, 22369621, 2251799724206762, 89478485, 36028796661050026, 357913941

Offset

0,2

References

Stephen Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

Links

Robert Price, Table of n, a(n) for n = 0..999

Eric Weisstein's World of Mathematics, Elementary Cellular Automaton

Index entries for sequences related to cellular automata

Index to Elementary Cellular Automata

Index entries for linear recurrences with constant coefficients, signature (0,21,0,-84,0,64).

Formula

From Colin Barker, Dec 27 2015 and Apr 14 2019: (Start)

a(n) = 21*a(n-2)-84*a(n-4)+64*a(n-6) for n>5.

G.f.: (1+2*x-16*x^2+64*x^3-96*x^5) / ((1-x)*(1+x)*(1-2*x)*(1+2*x)*(1-4*x)*(1+4*x)).

(End)

a(n) = 2*4^n - (8*2^n-1)/3 - 1 for odd n; a(n) = (4*2^n-1)/3 for even. - Karl V. Keller, Jr., Aug 27 2021

Mathematica

rule=13; rows=20; ca=CellularAutomaton[rule, {{1}, 0}, rows-1, {All, All}]; (* Start with single black cell *) catri=Table[Take[ca[[k]], {rows-k+1, rows+k-1}], {k, 1, rows}]; (* Truncated list of each row *) Table[FromDigits[catri[[k]], 2], {k, 1, rows}]   (* Decimal Representation of Rows *)

Prog

(Python) print([2*4**n - (8*2**n-1)//3 - 1 if n%2 else (4*2**n-1)//3 for n in range(50)]) # Karl V. Keller, Jr., Aug 27 2021

Crossrefs

Cf. A266282, A266283.

Sequence in context: A136106 A122696 A237267 * A215572 A023263 A070855

Adjacent sequences: A266281 A266282 A266283 * A266285 A266286 A266287

Keyword

nonn,easy

Author

Robert Price, Dec 26 2015

Status

approved