A071353 First term of the continued fraction expansion of (3/2)^n.
2, 4, 2, 16, 1, 2, 11, 1, 2, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 8, 5, 1, 7, 1, 25, 16, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 3, 1, 1, 2, 7, 4, 3, 2, 4, 1, 3, 1, 3, 1, 1, 1, 2, 10, 1, 2, 4, 1, 4, 2, 1, 3, 2, 14, 9, 6, 1, 11, 1, 1, 2, 1, 1, 2, 6, 1, 12, 1, 1, 2, 1, 2, 19, 12, 8, 1, 89, 59, 1, 3
Offset: 1
Examples
a(7) = 11 since floor(1/frac(3^7/2^7)) = floor(1/.0859375) = 11.
References
- S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 192-199.
Links
- G. C. Greubel, Table of n, a(n) for n = 1..1000
- Steven R. Finch, Powers of 3/2 Modulo One [From Steven Finch, Apr 20 2019]
- Steven R. Finch, Non-Ideal Waring's Problem [From Steven Finch, Apr 20 2019]
- Jeff Lagarias, 3x+1 Problem
- C. Pisot, La répartition modulo 1 et les nombres algébriques, Ann. Scuola Norm. Sup. Pisa, 7 (1938), 205-248.
- T. Vijayaraghavan, On the fractional parts of the powers of a number (I), J. London Math. Soc. 15 (1940) 159-160.
Programs
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Mathematica
Table[Floor[1/FractionalPart[(3/2)^n]], {n, 1, 100}] (* G. C. Greubel, Apr 18 2017 *) -
PARI
a(n) = contfrac((3/2)^n)[2] \\ Michel Marcus, Aug 01 2013
Formula
a(n) = floor(1/frac((3/2)^n)).
Comments