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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

Table of n, a(n) for n=0..5.

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

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