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

0,1

Links

Seiichi Manyama, Table of n, a(n) for n = 0..386

Wikipedia, Chebyshev polynomials.

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

Mathematica

Table[2*ChebyshevT[n, (n+1)/2], {n, 0, 18}] (* Amiram Eldar, Apr 09 2021 *)

Prog

(PARI) a(n) = 2*polchebyshev(n, 1, (n+1)/2);
(PARI) a(n) = round(2*cos(n*acos((n+1)/2)));
(PARI) a(n) = if(n==0, 2, 2*n*sum(k=0, n, (n-1)^k*binomial(n+k, 2*k)/(n+k)));

Crossrefs

Cf. A097691, A115066, A342205, A343259, A343261.

Sequence in context: A298440 A317808 A326909 * A246053 A345759 A062448

Adjacent sequences: A343257 A343258 A343259 * A343261 A343262 A343263

Keyword

nonn,easy

Author

Seiichi Manyama, Apr 09 2021

Status

approved