A049194 Number of digits in n-th term of A001387.
1, 2, 3, 6, 8, 12, 18, 27, 39, 58, 85, 125, 183, 269, 394, 578, 847, 1242, 1820, 2668, 3910, 5731, 8399, 12310, 18041, 26441, 38751, 56793, 83234, 121986, 178779, 262014, 384000, 562780, 824794, 1208795, 1771575, 2596370, 3805165, 5576741
Offset: 1
References
- Peter A. Hendriks, "A binary variant of Conway's audioactive sequence", lecture at 1192nd meeting of WWWW, Groningen, The Netherlands (Jul 15 1999).
Links
- Edward Krogius, Table of n, a(n) for n = 1..1000
- T. Sillke, The binary form of Conway's sequence
- Index entries for linear recurrences with constant coefficients, signature (1,1,0,0,-1).
Programs
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Mathematica
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 *) -
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
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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
Formula
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-1).
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.
From Jianing Song, Apr 28 2019: (Start)
a(n) = a(n-1) + a(n-2) - a(n-5) for n >= 7. [Derived from the T. Sillke link above.]
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).
Limit_{n->oo} a(n)/A001609(n) = c, where c = 1.276742... is the unique real root of 3x^3 - 4x^2 + x - 1. (End)
Extensions
More terms and formulas supplied by Gerton Lunter (gerton(AT)math.rug.nl)