cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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).

Original entry on oeis.org

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

Views

Author

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

Keywords

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.

Examples

			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, ...
		

Crossrefs

Cf. A060755, A305491 (numerators).

Programs

  • 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 *)

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

Extensions

More terms from Vladeta Jovovic, Apr 24 2001