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%I #67 Sep 05 2023 18:22:32
%S 0,1,1,5,7,47,37,319,533,1879,1627,20417,18107,263111,237371,52279,
%T 95549,1768477,1632341,33464927,155685007,166770367,156188887,
%U 3825136961,3602044091,19081066231,18051406831,57128792093,7751493599
%N Let u(1) = x and u(n+1) = (n^2/u(n)) + 1 for n >= 1; then a(n) is such that u(n) = (b(n)*x + c(n))/(a(n)*x + d(n)) (in lowest terms) and a(n), b(n), c(n), d(n) are positive integers.
%C For x real <> 1 - 1/log(2), Lim_{n -> infinity} abs(u(n)-n) = abs((x - 1)/(1 + (x - 1)*log(2))). [Corrected by _Petros Hadjicostas_, May 18 2020]
%C From _Petros Hadjicostas_, May 05 2020: (Start)
%C Given x > 0, u(n) = (A075827(n)*x + A075828(n))/(a(n)*x + A075829(n)) = (b(n)*x + c(n))/(a(n)*x + d(n)) with gcd(gcd(b(n), c(n)), gcd(a(n), d(n))) = 1 for each n >= 1.
%C Conjecture 1: Define the sequences (A(n): n >= 1) and (B(n): n >= 1) by A(n+1) = n^2/A(n) + 1 for n >= 2 with A(1) = infinity and A(2) = 1, and B(n+1) = n^2/B(n) + 1 for n >= 3 with B(1) = 0, B(2) = infinity, and B(3) = 1. Then a(n) = denominator(A(n)), b(n) = numerator(A(n)), c(n) = numerator(B(n)), and d(n) = denominator(B(n)) (assuming infinity = 1/0). Also, gcd(a(n), d(n)) = 1.
%C In 2002, _Michael Somos_ claimed that d(n) = A024168(n-1)/gcd(A024168(n-1), A024168(n)) for n >= 2. In 2006, _N. J. A. Sloane_ claimed that a(n) = A058313(n-1) for n >= 2 while _Alexander Adamchuk_ claimed that d(n) = A058312(n-1) - A058313(n-1) for n >= 2.
%C Conjecture 2: a(n) = A024167(n-1)/gcd(A024167(n-1), A024167(n)).
%C Conjecture 3: b(p) = a(p+1) for p = 1 or prime. In general, it seems that b(n) = A048671(n)*a(n+1) for all n for which A048671(n) < n.
%C Conjecture 4: c(n) = n*(a(n) + d(n)) - b(n) for n >= 1. (End)
%C All conjectures are proved in the link below except for the second part of Conjecture 3. - _Petros Hadjicostas_, May 21 2020
%H Petros Hadjicostas, <a href="/A075829/a075829.pdf">Proofs of various results about the sequence u(n)</a>, 2020.
%o (PARI) u(n)=if(n<2,x,(n-1)^2/u(n-1)+1);
%o a(n)=polcoeff(denominator(u(n)),1,x);
%Y Apart from the leading term, same as A058313.
%Y Cf. A075827 (= b), A075828 (= c), A075829 (= d).
%Y Cf. A024167, A024168, A048671, A058312.
%K nonn
%O 1,4
%A _Benoit Cloitre_, Oct 14 2002
%E Name edited by _Petros Hadjicostas_, May 04 2020
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