|
Search: id:A117726
|
|
|
| A117726 |
|
Moreno and Wagstaff's arithmetical function T(n). |
|
+0
3
|
|
| 2, 4, 4, 4, 8, 8, 4, 8, 10, 8, 12, 8, 8, 16, 8, 8, 16, 12, 12, 16, 16, 8, 12, 16, 10, 24, 16, 8, 24, 16, 12, 16, 16, 16, 24, 20, 8, 24, 16, 16, 32, 16, 12, 24, 24, 16, 20, 16, 18, 28, 24, 16, 24, 32, 16, 32, 16, 8, 36, 16, 24, 32, 20, 16, 32, 32, 12, 32, 32, 16, 28, 24, 16, 40, 28, 24
(list; graph; listen)
|
|
|
OFFSET
|
1,1
|
|
|
COMMENT
|
Also 4 times Kronecker's function F(n).
F(n) is the number of odd classes of binary quadratic forms ax^2+2bxy+cy^2 of discriminat b^2-ac = -n, where classes of the shape a(x^2+y^2) are counted as 1/2, and "odd" means that at least one of a and c is odd.
|
|
REFERENCES
|
L. Kronecker, Crelle, Vol. LVII (1860), p. 248; Werke, Vol. IV, p. 188.
C. J. Moreno and S. S. Wagstaff, Jr., Sums of Squares of Integers, Chapman and Hall, 2006, p. 43.
H. J. S. Smith, Report on the Theory of Numbers, reprinted in Vol. 1 of his Collected Math. Papers, Chelsea, NY, 1979, see pp. 323 (definition of F), 338 (g.f.).
|
|
FORMULA
|
G.f. for F(n): Sum_{n >= 1} F(n) q^n = (q^(1/4) / Sum_{ m=-infinity, infinity } q^( (2*m+1)^2/4 )) * Sum{ n=-infinity, infinity } q^(n^2+n-1)/(1-q^(2*n-1))^2.
|
|
MAPLE
|
t10:=add( q^( (2*m+1)^2/4 ), m=-20..20); t1:=series(q^(1/4)/t10, q, 100); t2:=add( q^(n^2+n-1)/(1-q^(2*n-1))^2, n=1..100): series(4*t1*t2, q, 100);
|
|
CROSSREFS
|
Cf. A117728, A005875.
Sequence in context: A043552 A035545 A023820 this_sequence A103228 A076340 A076345
Adjacent sequences: A117723 A117724 A117725 this_sequence A117727 A117728 A117729
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
njas, Apr 14 2006
|
|
|
Search completed in 0.002 seconds
|
