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A179897 Numbers a_n with property that a(n) is arithmetic mean of sequence "n-times n^-1, once n^(2*n+1)" that has integer valued both arithmetic and geometric means even though some of the sequence members are (for n>1) non-integer. 0
1, 11, 547, 52429, 8138021, 1865813431, 593445188743, 250199979298361, 135085171767299209, 90909090909090909091 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

According to search at OEIS for particular sequence members, a(n) is also: (1+2*n)-th q-integer for q=-n, (2*(n+1))-th cyclotomic polynomial at q=-n, Gaussian binomial coefficient [2*n+1, 2*n] for q=-n, number of walks of length 1+2*n between any two distinct vertices of the complete graph K_(n+1).

LINKS

Table of n, a(n) for n=1..10.

Google Groups, Integer-valued arithmetic and geometric means of sequences with non-integer numbers

FORMULA

a(n) = (n^(2*n+1) + 1) / (n+1). a(n) = \sum_{i=0}^{2*n} (-n)^i.

EXAMPLE

For n = 2 the a(2) = 11 since arithmetic mean of (1/2, 1/2, 2^5) is 33 / 3 = 11 where its geometric mean is 8^(1/3) = 2, i.e. both are integral.

PROG

(Python) [(n**(2*n+1)+1)//(n+1) for n in range(1, 11)]

CROSSREFS

Values for n = 5, 6 via other ways. Q-integers: A014986, A014987, K_n paths: A015531, A015540, Cyclotomic polynomials: A020504, A020505, Gaussian binomial coefficients: A015391, A015429.

Sequence in context: A065823 A233198 A049654 * A185203 A265978 A263184

Adjacent sequences:  A179894 A179895 A179896 * A179898 A179899 A179900

KEYWORD

easy,nonn

AUTHOR

Martin Saturka (martin(AT)saturka.net), Jul 31 2010

STATUS

approved

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Last modified September 21 07:28 EDT 2021. Contains 347596 sequences. (Running on oeis4.)