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A266665 Total number of OFF (white) cells after n iterations of the "Rule 47" elementary cellular automaton starting with a single ON (black) cell. 1
0, 1, 4, 6, 13, 15, 26, 28, 43, 45, 64, 66, 89, 91, 118, 120, 151, 153, 188, 190, 229, 231, 274, 276, 323, 325, 376, 378, 433, 435, 494, 496, 559, 561, 628, 630, 701, 703, 778, 780, 859, 861, 944, 946, 1033, 1035, 1126, 1128, 1223, 1225, 1324, 1326, 1429 (list; graph; refs; listen; history; text; internal format)
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..1000

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

Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

FORMULA

From Colin Barker, Jan 03 2016: (Start)

a(n) = (n^2+2*n+(-1)^n*(n-1)-1)/2 for n>0.

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

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

(End)

MAPLE

A266665:=n->(n^2+2*n+(-1)^n*(n-1)-1)/2: 0, seq(A266665(n), n=1..100); # Wesley Ivan Hurt, Jan 31 2017

MATHEMATICA

rule=47; 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 *)

Prepend[Table[(n^2 + 2 n + (-1)^n (n - 1) - 1)/2, {n, 52}] , 0] (* or *)

Table[SeriesCoefficient[x (1 + 3 x + x^3 - x^4)/((1 - x)^3 (1 + x)^2), {x, 0, n}], {n, 0, 52}] (* Michael De Vlieger, Jan 03 2016 *)

PROG

(PARI) concat(0, Vec(x*(1+3*x+x^3-x^4)/((1-x)^3*(1+x)^2) + O(x^100))) \\ Colin Barker, Jan 03 2016

CROSSREFS

Cf. A266659.

Sequence in context: A137821 A010061 A280557 * A249715 A249440 A063186

Adjacent sequences:  A266662 A266663 A266664 * A266666 A266667 A266668

KEYWORD

nonn,easy

AUTHOR

Robert Price, Jan 02 2016

STATUS

approved

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Last modified March 22 04:32 EDT 2019. Contains 321406 sequences. (Running on oeis4.)