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.
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).
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
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
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