OFFSET
0,2
COMMENTS
Initialized with a single black (ON) cell at stage zero.
Similar to A001045.
It is not difficult to prove that one has indeed a(n) = round(4*2^n/3) = A001045(n+2) for all n. The proof as well as the growth of the pattern is nearly identical to that of the toothpick sequence A139250. - M. F. Hasler, Feb 13 2020
The decimal representations of the n-th interval of elementary cellular automata rules 28 and 156 (see A266502 and A266508) generate this sequence. - Karl V. Keller, Jr., Sep 03 2021
LINKS
Robert Price, Table of n, a(n) for n = 0..126
Robert Price, Diagrams of first 20 stages
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
Stephen Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
Wolfram Research, Wolfram Atlas of Simple Programs
Index entries for linear recurrences with constant coefficients, signature (1,2).
FORMULA
From Colin Barker, Mar 14 2017: (Start)
G.f.: (1 + 2*x) / ((1 + x)*(1 - 2*x)).
a(n) = (2^(n+2) - 1) / 3 for n even.
a(n) = (2^(n+2) + 1) / 3 for n odd.
a(n) = a(n-1) + 2*a(n-2) for n>1.
(End)
I.e., a(n) = A001045(n+2) = A154917(n+2) = A167167(n+2) = |A077925(n+1)| = A328284(n+5) = round(4*2^n/3), cf. comments. - M. F. Hasler, Feb 13 2020
E.g.f.: (4*exp(2*x) - exp(-x))/3. - Stefano Spezia, Feb 13 2020
a(n) = floor((4*2^n + 1)/3). - Karl V. Keller, Jr., Sep 03 2021
MATHEMATICA
CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 678; stages = 128;
rule = IntegerDigits[code, 2, 10];
g = 2 * stages + 1; (* Maximum size of grid *)
a = PadLeft[{{1}}, {g, g}, 0, Floor[{g, g}/2]]; (* Initial ON cell on grid *)
ca = a;
ca = Table[ca = CAStep[rule, ca], {n, 1, stages + 1}];
PrependTo[ca, a];
(* Trim full grid to reflect growth by one cell at each stage *)
k = (Length[ca[[1]]] + 1)/2;
ca = Table[Table[Part[ca[[n]] [[j]], Range[k + 1 - n, k - 1 + n]], {j, k + 1 - n, k - 1 + n}], {n, 1, k}];
Table[FromDigits[Part[ca[[i]] [[i]], Range[i, 2 * i - 1]], 2], {i , 1, stages - 1}]
PROG
(Python) print([(4*2**n + 1)//3 for n in range(50)]) # Karl V. Keller, Jr., Sep 03 2021
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Robert Price, Mar 12 2017
STATUS
approved