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A071291 Second term of the continued fraction expansion of (3/2)^n; or 0 if no term is present. 2
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, 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, 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,9

COMMENTS

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

"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

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.

FORMULA

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

EXAMPLE

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

MATHEMATICA

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 *)

PROG

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

CROSSREFS

Cf. A006543, A071353.

Sequence in context: A241191 A352232 A221195 * A049330 A274040 A266363

Adjacent sequences:  A071288 A071289 A071290 * A071292 A071293 A071294

KEYWORD

easy,nonn

AUTHOR

Paul D. Hanna, Jun 10 2002

STATUS

approved

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Last modified August 12 07:45 EDT 2022. Contains 356067 sequences. (Running on oeis4.)