A319750 - OEIS

%I #16 Apr 11 2022 14:52:45

%S 1,3,33,3927,55602393,11147016454528647,

%T 448011292165037607943004375755833,

%U 723685043824607606355691108666081531638582859833105061571146291527

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

%C The numerators of the Heron sequence are in A319749.

%C 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).

%C n even: a(n) = A041019((5*2^n-5)/3).

%C n  odd: a(n) = A041019((5*2^n-1)/3).

%C 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.

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

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

%F h(n) = hn(n)/hd(n), hn(0) = 3, hd(0) = 1.

%F hn(n+1) = (hn(n)^2 + 13*hd(n)^2)/2.

%F hd(n+1) = hn(n)*hd(n).

%F A041018(n) = A010122(n)*A041018(n-1) + A041018(n-2).

%F A041019(n) = A010122(n)*A041019(n-1) + A041019(n-2).

%F 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

%e A078370(2) = 29.

%e hd(0) = A041047(0) = 1, hd(1) = A041047(3) = 5,

%e hd(2) = A041047(5) = 135, hd(3) = A041047(13) = 38145.

%p hn[0]:=3: hd[0]:=1:

%p for n from 1 to 6 do

%p   hn[n]:=(hn[n-1]^2+13*hd[n-1]^2)/2:

%p   hd[n]:=hn[n-1]*hd[n-1]:

%p   printf("%5d%40d%40d\n", n, hn[n], hd[n]):

%p end do:

%o (Python)

%o def aupton(nn):

%o     hn, hd, alst = 3, 1, [1]

%o     for n in range(nn):

%o         hn, hd = (hn**2 + 13*hd**2)//2, hn*hd

%o         alst.append(hd)

%o     return alst

%o print(aupton(7)) # _Michael S. Branicky_, Mar 15 2022

%Y Cf. A041018, A041019, A078370, A010122, A041047, A319749.

%K nonn,frac,easy

%O 0,2

%A _Paul Weisenhorn_, Sep 27 2018

%E a(5) corrected and terms a(6) and a(7) added by _Peter Bala_, Mar 15 2022