A071315 -id:A071315 - cp's OEIS frontend

A071316 Sum of terms of continued fraction expansion of frac((3/2)^n).

2, 4, 5, 16, 10, 10, 17, 13, 20, 74, 113, 32, 25, 76, 55, 31, 44, 86, 74, 46, 42, 100, 402, 115, 63, 71, 104, 143, 489, 346, 96, 78, 68, 87, 167, 196, 116, 95, 76, 123, 109, 108, 141, 176, 141, 133, 260, 1038, 4748, 5731, 1162, 285, 189, 248, 478, 399, 163, 154

Offset: 1

Paul D. Hanna, Jun 11 2002

nonn

What is the rate of growth of this sequence?

			a(3) = 5 since frac((3/2)^3) = [0;2,1,2] and a(3) = 2 + 1 + 2.

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

Alois P. Heinz, 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

Cf. A002379, A071315.

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

Name corrected by Sean A. Irvine, Jul 11 2024

A071915 Number of 1's in the continued fraction expansion of (3/2)^n.

Original entry on oeis.org

0, 0, 1, 0, 2, 3, 3, 6, 3, 5, 1, 2, 8, 2, 3, 5, 2, 3, 3, 6, 10, 8, 6, 4, 2, 3, 6, 5, 2, 9, 12, 7, 17, 10, 7, 9, 8, 10, 13, 13, 10, 12, 14, 9, 11, 10, 11, 6, 9, 5, 3, 13, 13, 19, 18, 13, 8, 12, 15, 14, 18, 7, 19, 19, 17, 15, 13, 14, 16, 13, 20, 16, 10, 20, 25, 17, 19, 14, 19, 14, 18, 22

Offset: 1

Benoit Cloitre, Jun 13 2002

It seems that lim n ->infinity a(n)/n = 0.2... << (log(4)-log(3))/log(2) = 0.415... the expected density of 1's (cf. measure theory of continued fraction).

			The continued fraction of (3/2)^24 is [16834, 8, 1, 10, 2, 25, 1, 3, 1, 1, 57, 6] which contains 4 "1's", hence a(24)=4.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A071337, A071316, A071315, A071529.

a[1] = 0; a[n_] := Count[ContinuedFraction[(3/2)^n], 1]; Array[a, 100] (* Amiram Eldar, Sep 05 2020 *)

for(n=1,200,s=contfrac(frac((3/2)^n)); print1(sum(i=1,length(s),if(1-component(s,i),0,1)),","))

cp's OEIS Frontend

A071316 Sum of terms of continued fraction expansion of frac((3/2)^n).

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