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.

1, 0, 1, 1, 5, 13, 23, 101, 307, 641, 893, 7303, 9613, 97249, 122989, 19793, 48595, 681971, 818107, 13093585, 77107553, 66022193, 76603673, 1529091919, 1752184789, 7690078169, 8719737569, 23184641107, 3721854001, 96460418429

Offset

1,5

Comments

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]

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

From Petros Hadjicostas, May 06 2020: (Start)

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).

b(n) = A024167(n)/gcd(A024167(n-1), A024167(n)).

c(n) = A024168(n)/gcd(A024168(n-1), A024168(n)).

d(n) = A024167(n-1)/gcd(A024167(n-1), A024167(n)).

b(n) + c(n) = n*(d(n) + a(n)).

u(n) = (A024167(n)*x + A024168(n))/(A024167(n-1)*x + A024168(n-1)). (End)

Links

Table of n, a(n) for n=1..30.

Petros Hadjicostas, Proofs of various results about the sequence u(n), 2020.

Formula

a(n) = A024168(n-1)/gcd(A024168(n-1), A024168(n)). - Michael Somos, Oct 29, 2002

From Alexander Adamchuk, Jul 22 2006: (Start)

a(n) = A058312(n-1) - A058313(n-1) for n > 1 with a(1) = 1.

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)

Mathematica

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 *)

Prog

(PARI) u(n) = if(n<2, x, (n-1)^2/u(n-1)+1);
a(n) = polcoeff(denominator(u(n)), 0, x);

Crossrefs

Cf. A075827 (= b), A075828 (= c), A075830 (= d).

Cf. A024167, A024168, A058312, A058313, A334958.

Sequence in context: A213483 A049833 A083800 * A119248 A114998 A340564

Adjacent sequences: A075826 A075827 A075828 * A075830 A075831 A075832

Keyword

nonn

Author

Benoit Cloitre, Oct 14 2002

Extensions

Name edited by Petros Hadjicostas, May 06 2020

Status

approved