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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. 7
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified May 14 19:53 EDT 2021. Contains 343903 sequences. (Running on oeis4.)