A278714 Denominators of (n-1)*(n-3)/(6*(2*n-1)), for n >= 1. Denominators of Dedekind sum s(2, 2*n-1).

1, 18, 1, 14, 27, 22, 13, 18, 17, 38, 63, 46, 5, 162, 29, 62, 99, 14, 37, 234, 41, 86, 27, 94, 49, 306, 53, 22, 171, 118, 61, 378, 13, 134, 207, 142, 73, 90, 77, 158, 243, 166, 17, 522, 89, 182, 279, 38, 97, 594, 101, 206, 63, 214, 109, 666, 113, 46, 351, 238

Offset

1,2

Comments

For the numerators see A278713, also for references and details.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Formula

a(n) = denominator((n-1)*(n-3)/(6*(2*n-1))) (in lowest terms), n >= 1.

a(n) = denominator(r(n)), with r(n) = s(2,2*n-1) where s(2,k) = Sum_{r=1..(k-1)} (r/k)*(2*r/k - floor(2*r/k)- 1/2), for odd k.

From Robert Israel, Dec 07 2016: (Start)

(2n+59) a(n) = (2n-1) a(n+30).

a(n) = 6(2n-1)/b(n) where

b(n) = 1 if n == 2, 14, 20, or 26 (mod 30)

b(n) = 2 if n == 5, 11, 17, or 29 (mod 30)

b(n) = 3 if n == 0, 4, 6, 10, 12, 16, 22, or 24 (mod 30)

b(n) = 5 if n == 8 (mod 30)

b(n) = 6 if n == 1, 7, 9, 15, 19, 21, 25, or 27 (mod 30)

b(n) = 10 if n == 23 (mod 30)

b(n) = 15 if n == 18 or 28 (mod 30)

b(n) = 30 if n == 3 or 13 (mod 30).

G.f.: x*(1+18*x+x^2+14*x^3+27*x^4+22*x^5+13*x^6+18*x^7+17*x^8+38*x^9+63*x^10+46*x^11

+5*x^12+162*x^13+29*x^14+62*x^15+99*x^16+14*x^17+37*x^18+234*x^19+41*x^20+86*x^21

+27*x^22+94*x^23+49*x^24+306*x^25+53*x^26+22*x^27+171*x^28+118*x^29+59*x^30

+342*x^31+11*x^32+106*x^33+153*x^34+98*x^35+47*x^36+54*x^37+43*x^38+82*x^39

+117*x^40+74*x^41+7*x^42+198*x^43+31*x^44+58*x^45+81*x^46+10*x^47+23*x^48

+126*x^49+19*x^50+34*x^51+9*x^52+26*x^53+11*x^54+54*x^55+7*x^56+2*x^57+9*x^58

+2*x^59)/(1-x^30)^2.

(End)

Maple

seq(denom((n-1)*(n-3)/(6*(2*n-1))), n=1..100); # Robert Israel, Dec 07 2016

Mathematica

Table[((n-1)(n-3))/(6(2n-1)), {n, 60}]//Denominator (* Harvey P. Dale, Feb 10 2019 *)

Prog

(PARI) a(n) = denominator((n-1)*(n-3)/(6*(2*n-1))) \\ Felix Fröhlich, Nov 28 2016
(Magma) [Denominator((n-1)*(n-3)/(6*(2*n-1))): n in [1..60]]; // Vincenzo Librandi, Dec 08 2016

Crossrefs

Cf. A278713.

Sequence in context: A152209 A040338 A040339 * A040340 A040341 A111872

Adjacent sequences: A278711 A278712 A278713 * A278715 A278716 A278717

Keyword

nonn,frac,easy

Author

Wolfdieter Lang, Nov 28 2016

Status

approved