A080513 a(n) = round(n/2) + 1 = ceiling(n/2) + 1 = floor((n+1)/2) + 1.

1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 14, 15, 15, 16, 16, 17, 17, 18, 18, 19, 19, 20, 20, 21, 21, 22, 22, 23, 23, 24, 24, 25, 25, 26, 26, 27, 27, 28, 28, 29, 29, 30, 30, 31, 31, 32, 32, 33, 33, 34, 34, 35, 35

Offset

0,2

Comments

Number of ON (black) cells in the n-th iteration of the "Rule 70" elementary cellular automaton starting with a single ON (black) cell.

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 (1,1,-1).

Formula

From Colin Barker, Jan 14 2016: (Start)

a(n) = (2*n-(-1)^n+5)/4.

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

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

(End)

a(n) = 1 + A110654(n). - Philippe Deléham, Nov 23 2016

a(n) = A008619(n+1) = A110654(n+2) = A110654(n)+1 = A004526(n+3) = A140106(n+5); a(n+2) = a(n) + 1 for all n >= 0. - M. F. Hasler, Feb 14 2019

a(n) = a(n-1)*a(n-2) - Sum_{i=0..n-3} a(i). - Marc Morgenegg, Oct 04 2019

Mathematica

rule=70; 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 *) Table[Total[catri[[k]]], {k, 1, rows}] (* Number of Black cells in stage n *)

Prog

(PARI) a(n) = (2*n-(-1)^n+5)/4 \\ Colin Barker, Jan 14 2016
(PARI) Vec((1+x-x^2)/((1-x)^2*(1+x)) + O(x^100)) \\ Colin Barker, Jan 14 2016
(PARI) A080513(n)=n\/2+1 \\ M. F. Hasler, Feb 14 2019

Crossrefs

Cf. A110654, A266843.

Sequence in context: A130472 A076938 A065033 * A004526 A140106 A123108

Adjacent sequences: A080510 A080511 A080512 * A080514 A080515 A080516

Keyword

nonn,easy

Author

Robert Price, Jan 04 2016

Extensions

Simpler definition from M. F. Hasler, Feb 14 2019

Status

approved