A319750 a(n) is the denominator of the Heron sequence with h(0) = 3.

1, 3, 33, 3927, 55602393, 11147016454528647, 448011292165037607943004375755833, 723685043824607606355691108666081531638582859833105061571146291527

Offset

0,2

Comments

The numerators of the Heron sequence are 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) = (2*n+1)^2 + 4 have the same relationship between the denominator of the Heron sequence and the denominator of the continued fraction convergent to 2*n+1.

sqrt(c(n)) has the continued fraction [2*n+1; n, 1, 1, n, 4*n+2].

hn(n)^2 - c(n)*hd(n)^2 = 4 for n > 1.

Links

Table of n, a(n) for n=0..7.

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).

a(0) = 1, a(1) = 3 and a(n) = 2*T(2^(n-2), 11/2)*a(n-1) for n >= 2, where T(n,x) denotes the n-th Chebyshev polynomial of the first kind. - Peter Bala, Mar 16 2022

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:

Prog

(Python)
def aupton(nn):
    hn, hd, alst = 3, 1, [1]
    for n in range(nn):
        hn, hd = (hn**2 + 13*hd**2)//2, hn*hd
        alst.append(hd)
    return alst
print(aupton(7)) # Michael S. Branicky, Mar 15 2022

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,frac,easy

Author

Paul Weisenhorn, Sep 27 2018

Extensions

a(5) corrected and terms a(6) and a(7) added by Peter Bala, Mar 15 2022

Status

approved