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A327344 9 + 8*a(n) appears in a congruence which determines representative parallel primitive binary quadratic forms for discriminant 9*m(n)^2 - 4 and representation -m(n)^2, where m(n) = A002559(n) (Markoff numbers). 1
0, 0, 39, 273, 1365, 333, 12870, 46410, 10878, 88218, 304668, 107559, 1576614, 2852889, 4144413, 13637988, 28406235, 53558505, 12085458, 92899170, 133886883, 34633998, 351194025, 1334488428, 1819412595, 410100933, 3041210445, 4333538430, 1118696184, 9146719764, 15150288153, 29675764248 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

See A327343 for the relevance of representative parallel primitive binary quadratic forms (rpapfs) for discriminant Disc(n) = 9*m(n)^2 - 4 and representation -m(n)^2 for the determination of ordered Markoff triples.

These rpapfs FPa(n) = [-m(n)^2, B(n), - C(n)] are determined by the solutions of the congruence z^2 - d(n) == 0 (mod (m(n)/2)^2), where d(n) = 9 + 8*a(n) if m(n) = A002559(n) is even, and the integer z(n) = 1 + 2*J(n) = B(n)/4 is from the set {1, 3, ..., 2*(m(n)/2)^2 - 1}. The members C(n) = 7 + (z^2 - d(n))/(m(n)/2)^2. In the odd m(n) case the congruence is B(n)^2 - d(n) == 0 (mod m(n)^2) , where d(n) = 9 + 8*a(n) and B(n) = 2*j(n) + 1 from the set {1, 3, ..., 2*m(n)^2 -1}. The member C(n) = 1 + (1/4)*(B(n)^2 - d(n))/m(n)^2. The different solutions are then FPa(n;i), for i = 1, 2, ..., #FPa(n), with #FPa(n) = A327343(n) = 2^A327342(n).

The d(n) sequence begins with {9, 9, 321, 2193, 10929, 2673, 102969, 371289, 87033, 705753,...}

LINKS

Table of n, a(n) for n=1..32.

FORMULA

a(n) = (d(n) - 9)/8 = 37*M(n)*(1 + 8*M(n)) with M(n) = A309376(n) = (m(n)/2 -1)/16 if m(n) is even, and a(n) = (d(n) - 9)/8 = 13*M(n)*(1 + 2*M(n)) with M(n) = A309376(n) = (m(n)-1)/4 if m(n) is odd.

EXAMPLE

n = 6: m(6)/2 = 17, M(6) = (17 - 1)/16 = 1, a(6) = 37*1*9 = 333. d(6) = 2673.

n = 7: m(7) = 89, M(7) = 22, a(7) = 13*22*45 = 12870. d(7) = 102969.

The two (#FPa(6) = 2^1) solutions z(6) = B(6)/4 are z(n;1) = 19 and z(n;2) = 1559. They lead to FPa(6;1) = [-34^2, 76, +1] and FPa(6;2) = [-34^2, 2236, -1079].

The two (#FPa(7) = 2^1) solutions B(7) are B(7;1) = 199 and B(7;2) = 15643 (the upper bound was 2*m(7)^2 - 1 = 15841), leading to FPa(7;1) = [-89^2, 199, +1] and FPa(7;2) = [-89^2, 15643, -7721].

In both cases the second solution leads to the ordered Markoff triples MT(6) = (1, 13, 34) and MT(7) = (1, 34, 89). The other solution leads to the unordered triples (1, 34, 13) and (1, 89, 34).

CROSSREFS

Cf. A002559, A309376, A327342, A327343.

Sequence in context: A101627 A229639 A070146 * A190537 A187095 A232874

Adjacent sequences:  A327341 A327342 A327343 * A327345 A327346 A327347

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Sep 11 2019

STATUS

approved

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Last modified November 28 01:24 EST 2021. Contains 349396 sequences. (Running on oeis4.)