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A173148
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a(n) = cos(2*n*arccos(sqrt(n))).
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9
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1, 1, 17, 485, 18817, 930249, 55989361, 3974443213, 325142092801, 30122754096401, 3117419602578001, 356452534779818421, 44627167107085622401, 6071840759403431812825, 892064955046043465408177, 140751338790698080509966749, 23737154316161495960243527681
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OFFSET
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0,3
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COMMENTS
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The Chebyshev polynomial T_n is defined by cos(nx) = T_n(cos(x)). So T_2n(cos(x)) = cos(2nx) = cos^2(nx) - 1 = (T_n(x))^2 - 1 consists of only even powers of x. As a result, a(n) = T_2n(sqrt(n)) is an integer. - Michael B. Porter, Jan 01 2019
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LINKS
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FORMULA
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a(n) = Sum_{k=0..n} binomial(2*n,2*k)*(n-1)^(n-k)*n^k. - Seiichi Manyama, Dec 27 2018
a(n) = T_{2*n}(sqrt(n)) = T_{n}(2*n-1) where T_{n}(x) is a Chebyshev polynomial of the first kind. - Seiichi Manyama, Dec 29 2018
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MATHEMATICA
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Table[Round[Cos[2 n ArcCos[Sqrt[n]]]], {n, 0, 30}] (* Artur Jasinski, Feb 11 2010 *)
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PROG
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(PARI) {a(n) = sum(k=0, n, binomial(2*n, 2*k)*(n-1)^(n-k)*n^k)} \\ Seiichi Manyama, Dec 27 2018
(PARI) {a(n) = round(cosh(2*n*acosh(sqrt(n))))} \\ Seiichi Manyama, Dec 27 2018
(GAP) a:=List([0..20], n->Sum([0..n], k->Binomial(2*n, 2*k)*(n-1)^(n-k)*n^k));; Print(a); # Muniru A Asiru, Jan 03 2019
(Magma) [&+[Binomial(2*n, 2*k)*(n-1)^(n-k)*n^k: k in [0..n]]: n in [0..20]]; // Vincenzo Librandi, Jan 03 2019
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CROSSREFS
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Cf. A053120 (Chebyshev polynomial), A132592, A146311, A146312, A146313, A173115, A173116, A173121, A173127, A173128, A173129, A173130, A173131, A173133, A173134, A322790.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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