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

1,2

References

Peter A. Hendriks, "A binary variant of Conway's audioactive sequence", lecture at 1192nd meeting of WWWW, Groningen, The Netherlands (Jul 15 1999).

Links

Table of n, a(n) for n=1..40.

T. Sillke, The binary form of Conway's sequence

Index entries for linear recurrences with constant coefficients, signature (1,1,0,0,-1).

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.

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).

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)

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

Cf. A001387, A001609.

Sequence in context: A325342 A085642 A270738 * A058298 A299758 A303703

Adjacent sequences: A049191 A049192 A049193 * A049195 A049196 A049197

Keyword

base,easy,nonn

Author

Olivier Gérard.

Extensions

More terms and formulas supplied by Gerton Lunter (gerton(AT)math.rug.nl)

Status

approved