A371639 a(n) = numerator(Voronoi(3, 2*n)) where Voronoi(c, n) = ((c^n - 1) * Bernoulli(n)) / (n * c^(n - 1)).

2, -2, 26, -82, 1342, -100886, 1195742, -57242642, 31945440878, -276741323122, 26497552755742, -9169807783193206, 418093081574417342, -66910282127782482482, 37158050152167281792026, -2626016090388858294953362, 632184834985453539204543742, -1543534415494449734887808117378

Offset: 1

Author

Peter Luschny, Mar 30 2024

Keywords

• sign
• frac

Comments

To begin with, we observe that if c = 2, then the numerator of Voronoi(2, 2*n) is the same as the numerator of Euler(2*n - 1, 1), which is equal to (-1)^n*A002425(n). Similarly, the denominator of Voronoi(2, 2*n) is A255932(n), which is equal to 2^A292608(n). The rational sequence r(n) = a(n) / A371640(n) examines the corresponding relationships in the case c = 3.

The function Voronoi, which is defined in the Name, was inspired by Voronoi's congruence. This congruence states that for any even integer k >= 2 and all positive coprime integers c, n: (c^k - 1)*N(k) == k*c^(k-1)*D(k)*Sum_{m=1..n-1} m^(k-1)* floor(m*c / n) mod n, where N(k) = numerator(Bernoulli(k)), D(k) = denominator( Bernoulli(k)) and gcd(N(k), D(k)) = 1.

Examples

			r(n) = 2/9, -2/81, 26/2187, -82/6561, 1342/59049, -100886/1594323, ...
		

References

• Emma Lehmer, On congruences involving Bernoulli numbers and the quotients of Fermat and Wilson, Ann. Math. 39 (1938), 350-360.
• Štefan Porubský, Further Congruences Involving Bernoulli Numbers, Journal of Number Theory 16, 87-94 (1983).
• Georgy Feodosevich Voronyi, On Bernoulli numbers, Comm. Charkou Math. Sot. 2, 129-148 (1890) (in Russian).

Links

• Digital Library of Mathematical Functions, Voronoi's congruence.

Crossrefs

Cf. A371640 (denominator), A371638.

Cf. A002425, A255932, A292608.

Programs

Maple

Voronoi := (a, k) -> ((a^k - 1) * bernoulli(k)) / (k * a^(k - 1)):
VoronoiList := (a, len) -> local k; [seq(Voronoi(a, 2*k), k = 1..len)]:
numer(VoronoiList(3, 18));

Formula

a(n) = Voronoi(3, 2*n) * 3^(2*n + valuation(n, 3)).