|
| |
|
|
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).
|
|
2
|
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
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
|
|
|
|
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
|
|
|
|
KEYWORD
|
nonn,easy
|
|
|
AUTHOR
|
Avi Peretz (njk(AT)netvision.net.il), Apr 21 2001
|
|
|
EXTENSIONS
|
|
|
|
STATUS
|
approved
|
| |
|
|
