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A075829 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))/(d(n)*x + a(n)) (in lowest terms) and a(n), b(n), c(n), d(n) are positive integers.

%I A075829 #39 Jul 02 2025 19:29:18
%S A075829 1,0,1,1,5,13,23,101,307,641,893,7303,9613,97249,122989,19793,48595,
%T A075829 681971,818107,13093585,77107553,66022193,76603673,1529091919,
%U A075829 1752184789,7690078169,8719737569,23184641107,3721854001,96460418429
%N A075829 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))/(d(n)*x + a(n)) (in lowest terms) and a(n), b(n), c(n), d(n) are positive integers.
%C A075829 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 A075829 Difference between the denominator and the numerator of the (n-1)-th alternating harmonic number Sum_{k=1..n-1} (-1)^(k+1)*1/k = A058313(n-1)/A058312(n-1). - _Alexander Adamchuk_, Jul 22 2006
%C A075829 From _Petros Hadjicostas_, May 06 2020: (Start)
%C A075829 Inspired by _Michael Somos_'s result below, we established the following formulas (valid for n >= 2). All the denominators in the first three formulas are equal to A334958(n).
%C A075829 b(n) = A024167(n)/gcd(A024167(n-1), A024167(n)).
%C A075829 c(n) = A024168(n)/gcd(A024168(n-1), A024168(n)).
%C A075829 d(n) = A024167(n-1)/gcd(A024167(n-1), A024167(n)).
%C A075829 b(n) + c(n) = n*(d(n) + a(n)).
%C A075829 u(n) = (A024167(n)*x + A024168(n))/(A024167(n-1)*x + A024168(n-1)). (End)
%H A075829 Petros Hadjicostas, <a href="/A075829/a075829.pdf">Proofs of various results about the sequence u(n)</a>, 2020.
%F A075829 a(n) = A024168(n-1)/gcd(A024168(n-1), A024168(n)). - _Michael Somos_, Oct 29 2002
%F A075829 From _Alexander Adamchuk_, Jul 22 2006: (Start)
%F A075829 a(n) = A058312(n-1) - A058313(n-1) for n > 1 with a(1) = 1.
%F A075829 a(n) = denominator(Sum_{k=1..n-1} (-1)^(k+1)*1/k) - numerator(Sum_{k=1..n-1}(-1)^(k+1)*1/k). (End)
%t A075829 Denominator[Table[Sum[(-1)^(k+1)*1/k,{k,1,n-1}],{n,1,30}]]-Numerator[Table[Sum[(-1)^(k+1)*1/k,{k,1,n-1}],{n,1,30}]] (* _Alexander Adamchuk_, Jul 22 2006 *)
%o A075829 (PARI) u(n) = if(n<2, x, (n-1)^2/u(n-1)+1);
%o A075829 a(n) = polcoeff(denominator(u(n)), 0, x);
%Y A075829 Cf. A075827 (= b), A075828 (= c), A075830 (= d).
%Y A075829 Cf. A024167, A024168, A058312, A058313, A334958.
%K A075829 nonn
%O A075829 1,5
%A A075829 _Benoit Cloitre_, Oct 14 2002
%E A075829 Name edited by _Petros Hadjicostas_, May 06 2020