A343260 a(n) = 2 * T(n,(n+1)/2) where T(n,x) is a Chebyshev polynomial of the first kind.
2, 2, 7, 52, 527, 6726, 103682, 1874888, 38925119, 912670090, 23855111399, 687808321212, 21687295069442, 742397047217294, 27420344506901023, 1086932029484351248, 46027034321342899967, 2073668380220713167378, 99042070146811639444802
Offset: 0
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..386
- Wikipedia, Chebyshev polynomials.
Programs
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Mathematica
Table[2*ChebyshevT[n, (n+1)/2], {n, 0, 18}] (* Amiram Eldar, Apr 09 2021 *) -
PARI
a(n) = 2*polchebyshev(n, 1, (n+1)/2);
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PARI
a(n) = round(2*cos(n*acos((n+1)/2)));
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PARI
a(n) = if(n==0, 2, 2*n*sum(k=0, n, (n-1)^k*binomial(n+k, 2*k)/(n+k)));
Formula
a(n) = 2 * cos(n*arccos((n+1)/2)).
a(n) = 2 * n * Sum_{k=0..n} (n-1)^k * binomial(n+k,2*k)/(n+k) for n > 0.
a(n) ~ exp(1) * n^n. - Vaclav Kotesovec, Apr 09 2021
Conjecture: a(p^r) == 1 (mod p^(2*r)) for positive integer r and all primes p >= 5. - Peter Bala, Mar 11 2024