OFFSET
0,1
COMMENTS
From Jianing Song, Dec 13 2025: (Start)
The Dirichlet character associated with the imaginary quadratic field Q(sqrt(-11)).
Note that (Sum_{i=0..10} i*a(i))/(-11) = 1 gives the class number of the imaginary quadratic field Q(sqrt(-11)), i.e., the corresponding ring of integers Z[(1+sqrt(-11))/2] is a unique factorization domain. (End)
REFERENCES
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 68.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Class Number.
Index entries for linear recurrences with constant coefficients, signature (-1,-1,-1,-1,-1,-1,-1,-1,-1,-1).
FORMULA
From Jianing Song, Dec 13 2025: (Start)
a(n) = (Prod_{1<=k<=5} sin(2*k*Pi/11))/(Prod_{1<=k<=5} sin(2*Pi/11)) = (sqrt(11)/2^5) * (Prod_{1<=k<=5} sin(2*k*Pi/11)).
Sum_{n>=1} a(n)/n = -(Pi/11^(3/2)) * (Sum_{i=0..10} i*a(i)) = Pi/sqrt(11) (Dirichlet class number formula). (End)
MATHEMATICA
Table[JacobiSymbol[n, 11], {n, 0, 80}] (* Jean-François Alcover, May 17 2017 *)
PROG
(PARI) A011582(n) = kronecker(n, 11);
for(n=0, 20, print1(A011582(n)", ")); /* R. J. Mathar, Feb 25 2012 */
CROSSREFS
Moebius transform of A035179.
Cf. A056874 (primes not inert in Q(sqrt(-11))), A296920 (primes decomposing), A191060 (primes remaining inert).
Kronecker symbols {(D/n)} for negative fundamental discriminants D = -3..-47, -67, -163: A102283, A101455, A175629, A188510, this sequence, A316569, A011585, A289741, A011586, A109017, A011588, A390614, A388073, A388072, A011591, A011592, A011596, A011615.
KEYWORD
sign,mult,easy
AUTHOR
STATUS
approved
