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A278715
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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.
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2
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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
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OFFSET
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2,11
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COMMENTS
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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.
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REFERENCES
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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.
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LINKS
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FORMULA
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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.
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EXAMPLE
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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.
...
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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.
...
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MATHEMATICA
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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 *)
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PROG
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(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
(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
(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
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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