

A080513


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


3



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
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OFFSET

0,2


COMMENTS

Number of ON (black) cells in the nth 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(n1)+a(n2)a(n3) for n>2.
G.f.: (1+xx^2) / ((1x)^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


MATHEMATICA

rule=70; rows=20; ca=CellularAutomaton[rule, {{1}, 0}, rows1, {All, All}]; (* Start with single black cell *) catri=Table[Take[ca[[k]], {rowsk+1, rows+k1}], {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+xx^2)/((1x)^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: A127365 A130472 A076938 * 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



