A255232 - OEIS

A255232

One half of the fundamental positive solution y = y1(n) of the first class of the Pell equation x^2 - 2*y^2 = -A007522(n), n >= 1 (primes congruent to 7 mod 8).

1, 2, 2, 3, 3, 4, 4, 4, 5, 6, 5, 5, 7, 6, 6, 7, 7, 9, 7, 7, 8, 10, 8, 9, 8, 8, 9, 11, 10, 9, 10, 13, 11, 10, 13, 14, 12, 11, 13, 11, 11, 12, 13, 12, 14, 13, 16, 12, 12, 17, 13, 14, 13, 16, 13, 18, 14, 16, 15, 14, 17, 14, 15, 14, 14, 14, 17, 16, 19, 16, 17, 16, 20, 21, 17, 16, 17, 16, 16

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OFFSET

1,2

COMMENTS

For the corresponding term x1(n) see A254938(n).

See A254938 also for the Nagell reference.

The least positive y solutions (that is those of the first class) for the primes +1 and -1 (mod 8) together (including prime 2) are given in A255246.

LINKS

M. F. Hasler, Table of n, a(n) for n = 1..1000, May 22 2025

FORMULA

A254938(n)^2 - 2*(2*a(n))^2 = -A007522(n) gives the smallest positive (proper) solution of this (generalized) Pell equation.

EXAMPLE

See A254938.

n = 3: 1^2 - 2*(2*2)^2 = 1 - 32 = -31 = -A007522(3).

PROG

(PARI) apply( {A255232(n, p=A007522(n))=Set(abs(qfbsolve(Qfb(-1, 0, 2), p, 1)))[1][2]\2}, [1..88]) \\ The 2nd optional arg allows to directly specify the prime. - M. F. Hasler, May 22 2025

CROSSREFS

Cf. A007522 (primes == 7 mod 8), A254938 (corresponding x1 values), A255233 (x2 values), A255234 (y2 values), A255246, A254935.

Sequence in context: A029122 A134482 A132921 * A181988 A389719 A194173

Adjacent sequences: A255229 A255230 A255231 * A255233 A255234 A255235

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Feb 18 2015

EXTENSIONS

More terms from Colin Barker, Feb 23 2015

Double-checked and extended by M. F. Hasler, May 22 2025

STATUS

approved