A266440 Total number of OFF (white) cells after n iterations of the "Rule 23" elementary cellular automaton starting with a single ON (black) cell.

0, 0, 5, 5, 14, 14, 27, 27, 44, 44, 65, 65, 90, 90, 119, 119, 152, 152, 189, 189, 230, 230, 275, 275, 324, 324, 377, 377, 434, 434, 495, 495, 560, 560, 629, 629, 702, 702, 779, 779, 860, 860, 945, 945, 1034, 1034, 1127, 1127, 1224, 1224, 1325, 1325, 1430

Offset

0,3

References

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

Links

Robert Price, Table of n, a(n) for n = 0..500

Eric Weisstein's World of Mathematics, Elementary Cellular Automaton

S. Wolfram, A New Kind of Science

Index entries for sequences related to cellular automata

Index to Elementary Cellular Automata

Formula

Conjectures from Colin Barker, Dec 30 2015 and Apr 15 2019: (Start)

a(n) = (n^2+((-1)^n+2)*n+(-1)^n-1)/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^2*(5-x^2) / ((1-x)^3*(1+x)^2).

(End)

Mathematica

rule=23; 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 *)

Crossrefs

Cf. A266434.

Sequence in context: A062367 A286257 A168418 * A317617 A294750 A304300

Adjacent sequences: A266437 A266438 A266439 * A266441 A266442 A266443

Keyword

nonn,easy

Author

Robert Price, Dec 29 2015

Status

approved