A266304 Total number of OFF (white) cells after n iterations of the "Rule 15" elementary cellular automaton starting with a single ON (black) cell.

0, 1, 5, 6, 14, 15, 27, 28, 44, 45, 65, 66, 90, 91, 119, 120, 152, 153, 189, 190, 230, 231, 275, 276, 324, 325, 377, 378, 434, 435, 495, 496, 560, 561, 629, 630, 702, 703, 779, 780, 860, 861, 945, 946, 1034, 1035, 1127, 1128, 1224, 1225, 1325, 1326, 1430

Offset: 0

Author

Robert Price, Dec 26 2015

Keywords

• nonn
• easy

References

• S. 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

Crossrefs

Cf. A266300.

Programs

Mathematica

rule=15; 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 *) nbc=Table[Total[catri[[k]]],{k,1,rows}]; (* Number of Black cells in stage n *) nwc=Table[Length[catri[[k]]]-nbc[[k]],{k,1,rows}]; (* Number of White cells in stage n *) Table[Total[Take[nwc,k]],{k,1,rows}] (* Number of White cells through stage n *)

Formula

Conjectures from Colin Barker, Dec 28 2015 and Apr 15 2019: (Start)

a(n) = n*(n+(-1)^n+2)/2.

a(n) = a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4)+a(n-5) for n>4.

G.f.: x*(1+4*x-x^2) / ((1-x)^3*(1+x)^2). (End)

a(n) = Sum_{i=1..n} (n-i-1) mod 2. - Wesley Ivan Hurt, Sep 15 2017