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A071047
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Number of 1's in n-th row of triangle in A071031, cellular automaton "rule 62".
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5
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1, 3, 3, 6, 5, 8, 9, 10, 10, 14, 13, 15, 16, 18, 18, 21, 20, 23, 24, 25, 25, 29, 28, 30, 31, 33, 33, 36, 35, 38, 39, 40, 40, 44, 43, 45, 46, 48, 48, 51, 50, 53, 54, 55, 55, 59, 58, 60, 61, 63, 63, 66, 65, 68, 69, 70, 70, 74, 73, 75, 76, 78, 78, 81, 80, 83, 84, 85
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OFFSET
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0,2
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COMMENTS
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Number of ON cells at generation n of 1-D CA defined by Rule 62. - N. J. A. Sloane, Aug 09 2014
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REFERENCES
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S. Wolfram, A New Kind of Science, Wolfram Media, 2002; Chapter 3.
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LINKS
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FORMULA
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Comments from N. J. A. Sloane, Aug 11 2014. (Start) As one can see from the illustration, there is a pattern that repeats every three steps on the left and every 12 steps on the right.
More precisely, let L(n) denote the number of N cells in the part of the diagram to the left of the central line. Then L(3t+1)=2t+1, L(3t+2)=L(3t+3)=2t+2, which is 0, 1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 8, 8, ... (essentially A004523). This has g.f. (x+x^2)/((1-x)(1-x^3)).
Let R(n) denote the number of ON cells on the central axis and to the right of this axis. Then R(0) through R(5) = [1, 2, 1, 4, 2, 4] and thereafter R(12k+6) through R(12k+17) = 7k + [5, 5, 4, 8, 6, 7, 8, 9, 8, 11, 9, 11] for k = 0,1,2,... The g.f. for R(n) is (1+2*x+x^2+3*x^3-x^4+x^5)/(1-x^3)(1-x^4)).
Combining these, we find that a(n) = L(n) + R(n) has the generating function that is given in the next line. (End)
G.f.: (2*x^5 + x^4 + 5*x^3 + 3*x^2 + 3*x + 1)/((1-x^3)*(1-x^4)). - Hans Havermann, May 26 2002
a(n+7) = a(n+4)+a(n+3)-a(n) with initial terms 1, 3, 3, 6, 5, 8, 9. - N. J. A. Sloane, Jan 31 2015
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MATHEMATICA
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CoefficientList[Series[(2 x^5 + x^4 + 5 x^3 + 3 x^2 + 3 x + 1)/((1 - x^3) (1 - x^4)), {x, 0, 80}], x] (* Vincenzo Librandi, Aug 10 2014 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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