OFFSET
0,2
LINKS
Robert Price, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
Stephen Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.
FORMULA
Conjectures from Colin Barker, Jan 02 2016 and Apr 16 2019: (Start)
a(n) = (31*2^n-4*((-1)^n+3))/24 for n>2.
a(n) = 2*a(n-1)+a(n-2)-2*a(n-3) for n>5. - [corrected by Karl V. Keller, Jr., Oct 07 2021]
G.f.: (1-x^4+x^5) / ((1-x)*(1+x)*(1-2*x)). (End)
Conjecture: a(n) = A000975(n) + 20*2^(n-5), for n>2. - Andres Cicuttin, Mar 31 2016
MAPLE
# Rule 41: value in generation r and column c, where c=0 is the central one
r41 := proc(r::integer, c::integer)
option remember;
local up ;
if r = 0 then
if c = 0 then
1;
else
0;
end if;
else
# previous 3 bits
[procname(r-1, c+1), procname(r-1, c), procname(r-1, c-1)] ;
up := op(3, %)+2*op(2, %)+4*op(1, %) ;
# rule 41 = 00101001_2: {5, 3, 0}->1, all others ->0
if up in {5, 3, 0} then
1;
else
0 ;
end if;
end if;
end proc:
A266613 := proc(n)
b := [seq(r41(r, 0), r=0..n)] ;
add(op(-i, b)*2^(i-1), i=1..nops(b)) ;
end proc:
smax := 20 ;
L := [seq(A266613(n), n=0..smax)] ; # R. J. Mathar, Apr 12 2019
MATHEMATICA
rule=41; 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 *) mc=Table[catri[[k]][[k]], {k, 1, rows}]; (* Keep only middle cell from each row *) Table[FromDigits[Take[mc, k], 2], {k, 1, rows}] (* Binary Representation of Middle Column *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Robert Price, Jan 01 2016
EXTENSIONS
Removed an unjustified claim that Colin Barker's conjectures are correct. Removed a program based on a conjecture. - N. J. A. Sloane, Jun 13 2022
STATUS
approved