A060723 a(n) is the denominator of r(n) where r(n) is the sequence of rational numbers defined by the recursion: r(0) = 0, r(1) = 1 and for n>1 r(n) = r(n-1) + r(n-2)/2. From this definition it is clear that a(n) is always a power of 2 (see A060755).

1, 1, 1, 2, 1, 4, 4, 8, 1, 16, 16, 32, 8, 64, 64, 128, 8, 256, 256, 512, 128, 1024, 1024, 2048, 256, 4096, 4096, 8192, 2048, 16384, 16384, 32768, 1024, 65536, 65536, 131072, 32768, 262144, 262144, 524288, 65536, 1048576, 1048576, 2097152

Offset

0,4

Comments

It can be proved that r(n) is an integer (i.e. a(n) = 1) if and only if n is one of 0, 1, 2, 4, 8.

Links

Table of n, a(n) for n=0..43.

Formula

r(n) = (((1/2)*(sqrt(3) + 1))^n - ((1/2)*(sqrt(3) - 1))^n * cos(Pi*n))/sqrt(3). - Peter Luschny, Jun 02 2018

Example

The sequence r(n) begins 0, 1, 1, 3/2, 2, 11/4, 15/4, 41/8, 7, 153/16, 209/16, 571/32, 363/16, 2023/64, 2749/64, 7521/128, 5135/64, ...

Mathematica

Denominator[RecurrenceTable[{a[0]==0, a[1]==1, a[n]==a[n-1]+a[n-2]/2}, a, {n, 50}]] (* Harvey P. Dale, Mar 07 2016 *)
Table[Denominator[Simplify[((1/2(1 + Sqrt[3]))^x - (1/2(Sqrt[3] - 1))^x Cos[Pi x])/ Sqrt[3]]], {x, 0, 43}]  (* Peter Luschny, Jun 02 2018 *)

Crossrefs

Cf. A060755, A305491 (numerators).

Sequence in context: A296188 A008312 A345442 * A300622 A195691 A074763

Adjacent sequences: A060720 A060721 A060722 * A060724 A060725 A060726

Keyword

nonn,easy

Author

Avi Peretz (njk(AT)netvision.net.il), Apr 21 2001

Extensions

More terms from Vladeta Jovovic, Apr 24 2001

Status

approved