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%I #31 May 10 2020 07:17:07
%S 1,2,3,6,8,12,18,27,39,58,85,125,183,269,394,578,847,1242,1820,2668,
%T 3910,5731,8399,12310,18041,26441,38751,56793,83234,121986,178779,
%U 262014,384000,562780,824794,1208795,1771575,2596370,3805165,5576741
%N Number of digits in n-th term of A001387.
%D Peter A. Hendriks, "A binary variant of Conway's audioactive sequence", lecture at 1192nd meeting of WWWW, Groningen, The Netherlands (Jul 15 1999).
%H T. Sillke, <a href="http://www.mathematik.uni-bielefeld.de/~sillke/SEQUENCES/series001">The binary form of Conway's sequence</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,0,0,-1).
%F a(n) = (8/9 + (1/18)*(748 - 36*sqrt(93))^(1/3) + (1/18)*(748 + 36*sqrt(93))^(1/3)) * (1/3 + (1/6)*(116 - 12*sqrt(93))^(1/3) + (1/6)*(116 + 12*sqrt(93))^(1/3))^n.
%F The number of digits is equal to c*l^n rounded down to the nearest integer, where c and l are the real roots of 3x^3 - 8x^2 + 5x - 1 and x^3 - x^2 - 1 respectively, for all n except n = 2 and n = 3.
%F From _Jianing Song_, Apr 28 2019: (Start)
%F a(n) = a(n-1) + a(n-2) - a(n-5) for n >= 7. [Derived from the T. Sillke link above.]
%F a(n) = a(n-1) + a(n-3) if n is odd, a(n-1) + a(n-3) + 1 if n is even, n >= 5 (this does not hold for n = 4).
%F Lim_{n->inf} a(n)/A001609(n) = c, where c = 1.276742... is the unique real root of 3x^3 - 4x^2 + x - 1. (End)
%t CoefficientList[Series[(1+x+x^3-x^4-x^5)/(1-x-x^2+x^5),{x,0,50}],x] (* _Peter J. C. Moses_, Jun 21 2013 *)
%o (PARI) a(n) = if (n==3, 3, if (n==4, 6, floor((8/9 + (1/18)*(748 - 36*sqrt(93))^(1/3) + (1/18)*(748 + 36*sqrt(93))^(1/3)) * (1/3 + (1/6)*(116 - 12*sqrt(93))^(1/3) + (1/6)*(116 + 12*sqrt(93))^(1/3))^(n-1)))) \\ _Michel Marcus_, Mar 04 2013
%o (PARI) a(n) = my(v=vector(n), u=[1,2,3,6]); if(n<=4, u[n], for(i=1, 4, v[i]=u[i]); for(i=5, n, v[i]=v[i-1]+v[i-3]+!(i%2)); v[n]) \\ _Jianing Song_, Apr 28 2019
%Y Cf. A001387, A001609.
%K base,easy,nonn
%O 1,2
%A _Olivier Gérard_.
%E More terms and formulas supplied by Gerton Lunter (gerton(AT)math.rug.nl)
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