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Irregular triangle read by rows: period lengths of periods of primitive Zagier-reduced binary quadratic forms with discriminants D(n) = A079896(n).
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%I #15 Nov 09 2016 15:03:47

%S 1,2,2,1,3,5,4,3,1,4,2,5,2,5,4,1,6,4,7,6,4,11,6,3,5,1,6,2,10,7,8,2,9,

%T 7,6,3,2,1,11,9,7,8,8,2,8,4,21,10,7,7,1,8,2,10,4,9,5,12,6

%N Irregular triangle read by rows: period lengths of periods of primitive Zagier-reduced binary quadratic forms with discriminants D(n) = A079896(n).

%C The possible positive nonsquare discriminants of binary quadratic forms are given in A079896.

%C For the definition of Zagier-reduced binary quadratic forms, see A257003.

%C A form is primitive if its coefficients are relatively prime.

%C The row sums give A257004(n), the number of primitive Zagier-reduced forms of discriminant D(n).

%C The number of entries in row n is A087048(n), the class number of primitive forms of discriminant D(n).

%D D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

%F a(n,k), n >= 0, k = 1, 2, ..., A079896(n), is the length of the k-th period of the primitive Zagier-reduced forms of discriminant D(n) = A079896(n). The lengths in row n are organized in nonincreasing order.

%e The table a(n,k) begins:

%e n/k 1 2 ... D(n) A087048(n) A257004(n)

%e 0: 1 5 1 1

%e 1: 2 8 1 2

%e 2: 2 1 12 2 3

%e 3: 3 13 1 3

%e 4: 5 17 1 5

%e 5: 4 20 1 4

%e 6: 3 1 21 2 4

%e 7: 4 2 24 2 6

%e 8: 5 2 28 2 7

%e 9: 5 29 1 5

%e 10: 4 1 32 2 5

%e 11: 6 4 33 2 10

%e 12: 7 37 1 7

%e 13: 6 4 40 2 10

%e 14: 11 41 1 11

%e 15: 6 3 44 2 9

%e 16: 5 1 45 2 6

%e 17: 6 2 48 2 8

%e 18: 10 52 1 10

%Y Cf. A257004, A257005, A079896, A225953, A079896.

%K nonn,tabf

%O 0,2

%A _Barry R. Smith_, Apr 20 2015