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A343260 a(n) = 2 * T(n,(n+1)/2) where T(n,x) is a Chebyshev polynomial of the first kind.

%I A343260 #22 Mar 15 2024 12:50:18
%S A343260 2,2,7,52,527,6726,103682,1874888,38925119,912670090,23855111399,
%T A343260 687808321212,21687295069442,742397047217294,27420344506901023,
%U A343260 1086932029484351248,46027034321342899967,2073668380220713167378,99042070146811639444802
%N A343260 a(n) = 2 * T(n,(n+1)/2) where T(n,x) is a Chebyshev polynomial of the first kind.
%H A343260 Seiichi Manyama, <a href="/A343260/b343260.txt">Table of n, a(n) for n = 0..386</a>
%H A343260 Wikipedia, <a href="https://en.wikipedia.org/wiki/Chebyshev_polynomials">Chebyshev polynomials</a>.
%F A343260 a(n) = 2 * cos(n*arccos((n+1)/2)).
%F A343260 a(n) = 2 * n * Sum_{k=0..n} (n-1)^k * binomial(n+k,2*k)/(n+k) for n > 0.
%F A343260 a(n) ~ exp(1) * n^n. - _Vaclav Kotesovec_, Apr 09 2021
%F A343260 Conjecture: a(p^r) == 1 (mod p^(2*r)) for positive integer r and all primes p >= 5. - _Peter Bala_, Mar 11 2024
%t A343260 Table[2*ChebyshevT[n, (n+1)/2], {n, 0, 18}] (* _Amiram Eldar_, Apr 09 2021 *)
%o A343260 (PARI) a(n) = 2*polchebyshev(n, 1, (n+1)/2);
%o A343260 (PARI) a(n) = round(2*cos(n*acos((n+1)/2)));
%o A343260 (PARI) a(n) = if(n==0, 2, 2*n*sum(k=0, n, (n-1)^k*binomial(n+k, 2*k)/(n+k)));
%Y A343260 Cf. A097691, A115066, A342205, A343259, A343261.
%K A343260 nonn,easy
%O A343260 0,1
%A A343260 _Seiichi Manyama_, Apr 09 2021