OFFSET
0,3
COMMENTS
a(n) is the arithmetic mean of the multiset consisting of n lots of 1/n and one lot of n^(2*n+1). This multiset also has an integer valued geometric mean which is equal to n for n > 0.
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
Andrew Howroyd, Table of n, a(n) for n = 0..100
FORMULA
a(n) = Sum_{i=0..2*n} (-n)^i.
EXAMPLE
For n = 2, a(2) = 11 which is the arithmetic mean of {1/2, 1/2, 2^5} = 33 / 3 = 11. The 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)]
(PARI) a(n) = (n^(2*n + 1) + 1)/(n + 1) \\ Andrew Howroyd, May 03 2023
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Martin Saturka (martin(AT)saturka.net), Jul 31 2010
EXTENSIONS
Edited, a(0)=1 prepended and more terms from Andrew Howroyd, May 03 2023
STATUS
approved