|
|
A257009
|
|
Number of finite sequences of positive integers with alternant equal to n.
|
|
6
|
|
|
4, 8, 9, 17, 14, 25, 22, 36, 25, 49, 31, 55, 49, 69, 41, 83, 52, 100, 66, 100, 66, 126, 84, 132, 88, 125, 95, 198, 82, 159, 119, 190, 125, 211, 125, 194, 135, 275, 128, 250, 152, 232, 191, 238, 174, 348, 150, 330, 223, 279, 158, 356, 220, 374, 217, 360, 196, 438
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
3,1
|
|
COMMENTS
|
The alternant of a sequence of positive integers (c_1, ..., c_r) with r>=3 is the positive integer [c_1, ..., c_r] - [c_2, ..., c_{r-1}], in which an expression in brackets denotes the numerator of the simplified rational number with continued fraction expansion having the sequence of quotients in brackets. The alternant of (c_1) is c_1 and the alternant of (c_1, c_2) is c_1*c_2. There are finitely many sequences with given alternant >= 3. (There are infinitely many sequences with alternant 2 -- (2), (1,2), (2,1), and all sequences of the form (1,p,1). It is for this reason that the offset is 3.)
The number of Zagier-reduced binary quadratic forms with discriminant equal to n^2-4 or n^2+4
The number of pairs of integers (h,k) with |k| < sqrt(D), k^2 congruent to D (mod 4), h > (sqrt(D) - k)/2, h exactly dividing (D-k^2)/4, where D=n^2+4 or n^2-4.
|
|
REFERENCES
|
D. B. Zagier, Zetafunktionen und quadratische Korper, Springer, 1981.
|
|
LINKS
|
|
|
FORMULA
|
a(n) equals the number of pairs (h,k) with |k| < sqrt(D), k^2 congruent to D (mod 4), h > (sqrt(D) - k)/2, h exactly dividing (D-k^2)/4, where D = n^2-4 or n^2+4.
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|