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A049194
Number of digits in n-th term of A001387.
5
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
1,2
REFERENCES
Peter A. Hendriks, "A binary variant of Conway's audioactive sequence", lecture at 1192nd meeting of WWWW, Groningen, The Netherlands (Jul 15 1999).
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)
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 *)
PROG
(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
(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
CROSSREFS
Sequence in context: A325342 A085642 A270738 * A058298 A299758 A303703
KEYWORD
base,easy,nonn
EXTENSIONS
More terms and formulas supplied by Gerton Lunter (gerton(AT)math.rug.nl)
STATUS
approved