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
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).
Programs
-
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 *) -
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
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
E.g.f.: ((2 + x)*cosh(x) + (3 + x)*sinh(x))/2. - Stefano Spezia, Aug 05 2025
Extensions
Simpler definition from M. F. Hasler, Feb 14 2019
Comments