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A319749 a(n) is the numerator of the Heron sequence with h(0)=3. 1
3, 11, 119, 14159, 200477279, 40191139395243839 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

The denominator of the Heron sequence is in A319750.

The following relationship holds between the numerator of the Heron sequence and the numerator of the continued fraction A041018(n)/A041019(n) convergent to sqrt(13).

n even: a(n)=A041018((5*2^n-5)/3).

n  odd: a(n)=A041018((5*2^n-1)/3).

More generally, all numbers c(n)=A078370(n)=(2n+1)^2+4 have the same relationship between the numerator of the Heron sequence and the numerator 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.

hn(0)=A041046(0)=5; hn(1)=A041046(3)=27; hn(2)=A041046(5)=727;

hn(3)=A041046(13)=528527.

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, A319750, A041046.

Sequence in context: A068693 A036930 A198085 * A209107 A015047 A339326

Adjacent sequences:  A319746 A319747 A319748 * A319750 A319751 A319752

KEYWORD

nonn

AUTHOR

Paul Weisenhorn, Sep 27 2018

STATUS

approved

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Last modified October 17 17:21 EDT 2021. Contains 348065 sequences. (Running on oeis4.)