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A179897 a(n) = (n^(2*n+1) + 1) / (n+1).

%I A179897 #12 May 03 2023 23:29:01
%S A179897 1,1,11,547,52429,8138021,1865813431,593445188743,250199979298361,
%T A179897 135085171767299209,90909090909090909091,74619186937936447687211,
%U A179897 73381705110822317661638341,85180949465178001182799643437,115244915978498073437814463065839,179766618030828831251710653305053711
%N A179897 a(n) = (n^(2*n+1) + 1) / (n+1).
%C A179897 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.
%C A179897 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).
%H A179897 Andrew Howroyd, <a href="/A179897/b179897.txt">Table of n, a(n) for n = 0..100</a>
%H A179897 Google Groups, <a href="http://groups.google.com/group/sci.math.research/msg/980fb8b3e847942e">Integer-valued arithmetic and geometric means of sequences with non-integer numbers</a>
%F A179897 a(n) = Sum_{i=0..2*n} (-n)^i.
%e A179897 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.
%o A179897 (Python) [(n**(2*n+1)+1)//(n+1) for n in range(1,11)]
%o A179897 (PARI) a(n) = (n^(2*n + 1) + 1)/(n + 1) \\ _Andrew Howroyd_, May 03 2023
%Y A179897 Main diagonal of A362783.
%Y A179897 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.
%K A179897 easy,nonn
%O A179897 0,3
%A A179897 Martin Saturka (martin(AT)saturka.net), Jul 31 2010
%E A179897 Edited, a(0)=1 prepended and more terms from _Andrew Howroyd_, May 03 2023