A266248 Decimal representation of the middle column of the "Rule 9" elementary cellular automaton starting with a single ON (black) cell.

1, 2, 5, 10, 21, 43, 86, 173, 346, 693, 1386, 2773, 5546, 11093, 22186, 44373, 88746, 177493, 354986, 709973, 1419946, 2839893, 5679786, 11359573, 22719146, 45438293, 90876586, 181753173, 363506346, 727012693, 1454025386, 2908050773, 5816101546, 11632203093

Offset: 0

Author

Robert Price, Dec 25 2015

Keywords

• nonn
• easy

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 (2,1,-2).

Crossrefs

Cf. A266243, A266247.

Programs

Mathematica

rule=9; 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 *)

Python

print([65*2**n//48 for n in range(50)]) # Karl V. Keller, Jr., Dec 15 2021

Formula

From Colin Barker, Dec 28 2015 and Apr 14 2019: (Start)

a(n) = (65*2^n-8*((-1)^n+3))/48 for n>3.

a(n) = 2*a(n-1)+a(n-2)-2*a(n-3) for n>6.

G.f.: (1+x^5-x^6) / ((1-x)*(1+x)*(1-2*x)).

(End)

a(n) = floor(65*2^n/48). - Karl V. Keller, Jr., Dec 15 2021