The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A319750 a(n) is the denominator of the Heron sequence with h(0)=3. 1
 1, 3, 33, 3927, 55602393, 11147216454528647 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS The numerator of the Heron sequence is in A319749. There is the following relationship between the denominator of the Heron sequence and the denominator of the continued fraction A041018(n)/A041019(n) convergent to sqrt(13). n even: a(n)=A041019((5*2^n-5)/3). n  odd: a(n)=A041019((5*2^n-1)/3). General: all numbers c(n)=A078370(n)=(2n+1)^2+4 have the same relationship between the denominator of the Heron sequence and the denominator of the continued fraction convergent to 2n+1. sqrt(c(n)) has the continued fraction 2n+1; n,1,1,n,4n+2. hn(n)^2-c(n)*hd(n)^2=4 for n>1. LINKS FORMULA h(n) = hn(n)/hd(n), hn(0)=3, hd(0)=1. hn(n+1) = (hn(n)^2+13*hd(n)^2)/2. hd(n+1) = hn(n)*hd(n). A041018(n) = A010122(n)*A041018(n-1) + A041018(n-2). A041019(n) = A010122(n)*A041019(n-1) + A041019(n-2). EXAMPLE A078370(2)=29. hd(0)=A041047(0)=1, hd(1)=A041047(3)=5, hd(2)=A041047(5)=135, hd(3)=A041047(13)=38145. MAPLE hn[0]:=3: hd[0]:=1: for n from 1 to 6 do   hn[n]:=(hn[n-1]^2+13*hd[n-1]^2)/2:   hd[n]:=hn[n-1]*hd[n-1]:   printf("%5d%40d%40d\n", n, hn[n], hd[n]): end do: CROSSREFS Cf. A041018, A041019, A078370, A010122, A041047, A319749. Sequence in context: A194889 A126675 A038694 * A204687 A134477 A080985 Adjacent sequences:  A319747 A319748 A319749 * A319751 A319752 A319753 KEYWORD nonn AUTHOR Paul Weisenhorn, Sep 27 2018 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified September 27 12:01 EDT 2020. Contains 337380 sequences. (Running on oeis4.)