A071291 - OEIS

%I #18 Apr 21 2019 09:34:53

%S 0,0,1,0,1,1,1,1,3,1,101,2,1,13,8,5,1,8,5,1,7,4,2,1,1,3,1,2,3,1,1,7,4,

%T 2,1,2,1,11,7,8,12,2,1,6,4,30,19,2,129,1,8,13,2,5,1,7,5,32,21,13,1,14,

%U 1,8,1,4,2,1,1,1,1,1,1,1,2,3,1,18,2,1,20,3,1,2,1,1,12,1,1,1,2,1,1,2,1

%N Second term of the continued fraction expansion of (3/2)^n; or 0 if no term is present.

%C What is the average of the reciprocal terms of this sequence?

%C "Pisot and Vijayaraghavan proved that frac((3/2)^n) has infinitely many accumulation points, i.e. infinitely many convergent subsequences with distinct limits. The sequence is believed to be uniformly distributed, but no one has even proved that it is dense in [0,1]." - S. R. Finch

%D S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 192-199.

%H G. C. Greubel, <a href="/A071291/b071291.txt">Table of n, a(n) for n = 1..1000</a>

%H Steven R. Finch, <a href="/FinchPwrs32.html">Powers of 3/2 Modulo One</a> [From Steven Finch, Apr 20 2019]

%H Steven R. Finch, <a href="/FinchWaring.html">Non-Ideal Waring's Problem</a> [From Steven Finch, Apr 20 2019]

%H Jeff Lagarias, <a href="http://www.cecm.sfu.ca/organics/papers/lagarias/index.html">3x+1 Problem</a>

%H C. Pisot, <a href="http://www.numdam.org/item?id=ASNSP_1938_2_7_3-4_205_0">La répartition modulo 1 et les nombres algébriques</a>, Ann. Scuola Norm. Sup. Pisa, 7 (1938), 205-248.

%H T. Vijayaraghavan, <a href="https://doi.org/10.1112/jlms/s1-15.2.159">On the fractional parts of the powers of a number (I)</a>, J. London Math. Soc. 15 (1940) 159-160.

%F a(n) = floor(1/frac(1/frac((3/2)^n))) when frac(1/frac((3/2)^n)) > 0; a(n) = 0 otherwise.

%e a(9) = 3 since floor(1/frac(1/frac(3^9/2^9))) = floor(1/frac(1/.443359375)) = 3.

%t a[n_] := If[FractionalPart[1/FractionalPart[(3/2)^n]] > 0, Floor[1/FractionalPart[1/FractionalPart[(3/2)^n]]], 0]; Table[a[n], {n, 1, 100}] (* _G. C. Greubel_, Apr 18 2017 *)

%o (PARI) a(n) = {cf = contfrac((3/2)^n); if (#cf < 3, return (0), return (cf[3]));} \\_Michel Marcus_, Aug 01 2013

%Y Cf. A006543, A071353.

%K easy,nonn

%O 1,9

%A _Paul D. Hanna_, Jun 10 2002