cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

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

Original entry on oeis.org

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

Views

Author

Wolfdieter Lang, Nov 28 2016

Keywords

Comments

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

Links

Crossrefs

Cf. A278713.

Programs

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

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)

A278715 Table T read by rows. T(k, h) gives the numerators of the Dedekind sums s(h, k) with gcd(h, k) = 1 or 0 otherwise.

Original entry on oeis.org

0, 1, -1, 1, 0, -1, 1, 0, 0, -1, 5, 0, 0, 0, -5, 5, 1, -1, 1, -1, -5, 7, 0, 1, 0, -1, 0, -7, 14, 4, 0, -4, 4, 0, -4, -14, 3, 0, 0, 0, 0, 0, 0, 0, -3, 15, 5, 3, 3, -5, 5, -3, -3, -5, -15, 55, 0, 0, 0, -1, 0, 1, 0, 0, 0, -55, 11, 4, 1, -1, 0, -4, 4, 0, 1, -1, -4, -11, 13, 0, 3, 0, 3, 0, 0, 0, -3, 0, -3, 0, -13, 91, 7, 0, 19, 0, 0, -7, 7, 0, 0, -19, 0, -7, -91
Offset: 2

Views

Author

Wolfdieter Lang, Nov 28 2016

Keywords

Comments

For the denominators see A278716.
The Dedekind sums are s(h,k) = Sum_{r=1..k-1} (r/k)*(h*r/k - floor(h*r/k) - 1/2) = Sum_{r=1..k-1} ((r/k))*((h*r)/k) with the period 1 sawtooth function ((x)) = x - floor(x) - 1/2 if x is not an integer and 0 otherwise. One assumes gcd(h,k) = 1, and for other h, k values the table has 0's. See the references Apostol pp. 52, 61-69, 72-73 and Ayoub pp. 168, 191. See also the Weisstein link.
In order to have a regular triangle T(k, h) one starts with k >= 2 and h = 1, 2, ..., k-1. s(1,1) = 0.

Examples

			The triangle T(k,h) begins (if gcd(k,h) is not 1 we use o instead of 0):
k\h  1  2  3  4  5  6  7   8  9  10  11  12
2:   0
3:   1 -1
4:   1  o -1
5:   1  0  0 -1
6:   5  o  o  o -5
7:   5  1 -1  1 -1 -5
8:   7  o  1  o -1  o -7
9:  14  4  o -4  4  o -4 -14
10:  3  o  0  o  o  o  0   o -3
11: 15  5  3  3 -5  5 -3  -3 -5 -15
12: 55  o  o  o -1  o  1   o  o   o -55
13: 11  4  1 -1  0 -4  4   0  1  -1  -4 -11
...
n = 14: 13 o 3 o 3 o o o -3 o -3 o -13,
n = 15: 91 7 0 19 0 0 -7 7 0 0 -19 0 -7 -91.
...
---------------------------------------------
The rational triangle s(h,k) begins (here o is used if gcd(h,k) is not 1):
k\h  1      2     3    4     5     6     7
2:   0
3:  1/18 -1/18
4:  1/8     o  -1/8
5:  1/5     0    0   -1/5
6:  5/18    o    o     o   -5/18
7:  5/14  1/14 -1/14  1/14 -1/14 -5/14
8:  7/16    o   1/16   o   -1/16   o   -7/16
...
n = 9: 14/27 4/27 o -4/27 4/27 o -4/27 -14/27,
n = 10: 3/5 o 0 o o o 0 o -3/5,
n = 11: 15/22 5/22 3/22 3/22 -5/22 5/22 -3/22 -3/22 -5/22 -15/22,
n = 12:  55/72 o o o -1/72 o 1/72 o o o  -55/72,
n = 13: 11/13 4/13 1/13 -1/13 0 -4/13 4/13 0 1/13 -1/13 -4/13 -11/13,
n = 14: 13/14 o 3/14 o 3/14 o o o -3/14 o -3/14 o -13/14,
n = 15: 1/90 7/18 o 19/90 o o -7/18 7/18 o o -19/90 o -7/18 -91/90.
...
--------------------------------------------

References

  • Apostol, Tom, M., Modular Functions and Dirichlet Series in Number Theory, Second edition, Springer, 1990.
  • Ayoub, R., An Introduction to the Analytic Theory of Numbers, Amer. Math. Soc., 1963.

Links

Crossrefs

Cf. A278716, A264388/A264389 (h=1), A278713/A278714 (h=2 odd k).

Programs

  • Magma
    [[GCD(n,k) eq 1 select Numerator((&+[(j/n)*(k*j/n - Floor(k*j/n) - 1/2): j in [1..(n-1)]])) else 0: k in [1..(n-1)]]: n in [2..15]]; // G. C. Greubel, Nov 22 2018
  • Mathematica
    T[n_, k_]:= If[GCD[n, k] == 1, Sum[(j/n)*(k*j/n - Floor[k*j/n] - 1/2), {j, 1, n - 1}], 0]; Numerator[Table[T[n, k], {n,2,15}, {k,1,n-1}]] //Flatten (* G. C. Greubel, Nov 22 2018 *)
  • PARI
    {T(n,k) = if(gcd(n,k)==1, sum(j=1,n-1, (j/n)*(k*j/n - floor(k*j/n) - 1/2)), 0)};
for(n=2,15, for(k=1,n-1, print1(numerator(T(n,k)), ", "))) \\ G. C. Greubel, Nov 22 2018
  • Sage
    def T(n,k):
    if gcd(n,k)==1:
       return numerator(sum((j/n)*(k*j/n - floor(k*j/n) - 1/2) for j in (1..(n-1))))
    elif gcd(n,k)!=1:
        return 0
    else:
        0
[[T(n,k) for k in (1..n-1)] for n in (2..15)] # G. C. Greubel, Nov 22 2018

Formula

T(k ,h) = numerator(s(h,k)) with the Dedekind sums s(h,k) given in a comment above and gcd(h,k) = 1. k >=2, h = 1, 2, ..., k-1. If gcd(h,k) is not 1 then T(k,h) is put to 0 (in the example o is used). Note that T(k,h) can vanish also for gcd(h,k) = 1.
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.