A389355 - OEIS

A389355

a(n) = (A388291(n)^2 - 1)/24.

0, 1, 7, 35, 330, 1190, 2262, 7812, 40426, 73151, 106267, 349692, 728365, 1373295, 2382030, 3432997, 9004975, 34217652, 46651605, 77979755, 111116370, 234531276, 388468927, 760917032, 1584781276, 3618302051, 7575315805, 10391390352, 11017977685, 35710034801

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OFFSET

1,3

COMMENTS

This sequence appears in the discriminants of the reduced principal Markoff forms F_p(X, Y) with odd Markoff numbers.

See A389354 for the definitions of Disc(n)= A305312(n), D(n) = A308687(n), the reduced principal indefinte binary quadratic form F_p(n; X, Y), for n >= 1, and the Frobenius - Markoff conjecture.

For odd Markoff numbers A388291(n) = A002559(A388292(n)) one has DiscOdd(n) = Disc(A388292(n)) = 1 + 4*D(A388292(n)).

The sequence member a(n) appears in the formula for DiscOdd(n) = 5 + (2*3)^3*a(n) = 5 + 216*a(n), for n >= 1.

LINKS

Table of n, a(n) for n=1..30.

FORMULA

a(n) = (A388291(n)^2 - 1)/24.

a(n) = Mo(n)*(1 + 2*Mo(n))/3, where Mo(n) = A309376(A366292(n)), for n >= 1.

EXAMPLE

a(0) = (1^2 - 1)/24 = 0.

a(2) = (5^2 - 1)/24 = 1.

a(2) = 1*(1 + 2*1)/3 = 1.

a(3) = (13^2 - 1)/24 = 168/24 = 7.