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 A319750 a(n) is the denominator of the Heron sequence with h(0) = 3. 1
 1, 3, 33, 3927, 55602393, 11147016454528647, 448011292165037607943004375755833, 723685043824607606355691108666081531638582859833105061571146291527 (list; graph; refs; listen; history; text; internal format)
 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 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:=3: hd:=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,      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

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Last modified May 20 09:27 EDT 2022. Contains 353871 sequences. (Running on oeis4.)