A262025 a(n) = (A262024(n)-1)/2: a(n)*(a(n) + 1) = d(n)*Y(n)^2 with d(n) = A007969 and Y(n) = A261250(n).

1, 4, 2, 9, 3, 324, 7, 16, 8, 4, 27, 98, 25, 63, 4900, 5, 11, 17, 36, 18, 12, 1024, 6, 99, 80, 12167, 49, 324, 33124, 242, 44, 7, 75, 9801, 15, 883159524, 31, 64, 32, 16, 3887, 125, 8, 1140624, 1849, 28899, 175, 26, 81, 27, 142884, 5202, 250000, 9, 575, 6075, 1071647, 19, 31404816, 49, 100, 50, 20, 16040025, 675, 79035335993124, 10, 147, 63, 602176, 512, 4900, 324, 153458

Offset: 1

Author

Wolfdieter Lang, Sep 19 2015

Keywords

• nonn

Comments

The positive fundamental solutions (x0(n), y0(n)) of the Pell equation x^2 - d(n) y^2 = +1, with d not a square, have only even y solutions for d(n) = A007969 (Conway's products of 1-happy couples). The proof is now given in the W. Lang link under A007969. The solutions x0 and y0 = 2*Y0 are given in A262024 and 2*A261250, respectively. The numbers X0(n) = (x0(n) - 1)/2 = a(n) satisfy a(n)*(a(n) + 1) = d(n)*Y0(n)^2. See the mentioned link.

Crossrefs

Cf. A007969, A007970, A262024, A261250.