A382209 -id:A382209 - cp's OEIS frontend

A383734 Numbers k such that 2+k and 2*k are squares.

2, 98, 3362, 114242, 3880898, 131836322, 4478554082, 152139002498, 5168247530882, 175568277047522, 5964153172084898, 202605639573839042, 6882627592338442562, 233806732499933208098, 7942546277405390632802, 269812766699283348307202, 9165691521498228451812098

Offset: 1

Emilio Martín, May 07 2025

The limit of a(n+1)/a(n) is 33.97056... = 17+12*sqrt(2) = (3+2*sqrt(2))^2 (see A156164).

			98 is a term becouse 98+2=100 is a square and 98*2=196 is a square.

Eric Weisstein's World of Mathematics, NSW numbers.
Index entries for linear recurrences with constant coefficients, signature (35,-35,1).

Cf. A002315, A088165, A008843, A008844, A075870, A031396, A156164.
Cf. A382209 (10+k and 10*k are squares).
Cf. A245226 (m such that k+m and k*m are squares).

LinearRecurrence[{35, -35, 1}, {2, 98, 3362}, 20] (* Amiram Eldar, May 07 2025 *)

from itertools import islice
def A383734_gen(): # generator of terms
    x, y = 1, 7
    while True:
        yield 2*x**2
        x, y = y, 6*y - x
A383734_list = list(islice(A383734_gen(), 100))

a(n) = (1/2) * ((3+2*sqrt(2))^(2*n-1) + (3-2*sqrt(2))^(1-2*n)) - 1.
a(n) = -2*sqrt(2)*sinh(n*log(17+12*sqrt(2))) + 3*cosh(n*log(17+12*sqrt(2))) - 1.
a(n) = 2*A002315(n-1)^2.
a(n) = A075870(n)^2 - 2.
a(n) = 34*a(n-1) - a(n-2) + 32.
G.f.: 2 * (1 + 14*x + x^2) / ((1 - x)*(1 - 34*x + x^2)). - Stefano Spezia, May 08 2025

A383898 a(n) is the smallest nonnegative integer k such that n + k and n*k are squares, or -1 if there is no such number.

Original entry on oeis.org

0, 2, -1, 0, 20, -1, -1, 8, 0, 90, -1, -1, 4212, -1, -1, 0, 272, 18, -1, 5, -1, -1, -1, -1, 0, 650, -1, -1, 142100, -1, -1, 32, -1, -1, -1, 0, 1332, -1, -1, 360, 41984, -1, -1, -1, 180, -1, -1, -1, 0, 50, -1, 117, 1755572, -1, -1, -1, -1, 568458, -1, -1, 53872730964

Offset: 1

Gonzalo Martínez, May 15 2025

sign

a(m^2) = 0, for all positive integers m.
If m is not in A245226 then a(m) = -1. Indeed, if (u, v) is the smallest solution of the equation x^2 - m*y^2 = m, then m + m*v^2 = u^2 and m*(m*v^2) = (m*v)^2. Therefore, a(m) = m*v^2.
The sequence A383734 contains the numbers k such that 2 + k and 2*k are squares, where A383734(2) = a(2). Similarly, A382209(1) = a(10).
a(181) <=  223502910856088814900 = 181*1111225770^2. - Michel Marcus, May 24 2025

			If n = 5, then 20 satisfies that 5 + 20 = 5^2 and 5*20 = 10^2, where 20 is the smallest integer greater than 5 with this property. So, a(5) = 20.

Michel Marcus, Table of n, a(n) for n = 1..180

Cf. A382209, A383734, A245226.

isok(m) = if (issquare(4*m), 1, #qfbsolve(Qfb(1, 0, -m), m, 2)); \\ A245226
a(n) = if (!isok(n), return(-1)); my(k=0); while (!(issquare(n+k) && issquare(n*k)), k++); k; \\ Michel Marcus, May 21 2025

isok(m) = if (issquare(4*m), 1, #qfbsolve(Qfb(1, 0, -m), m, 2)); \\ A245226
a(n) = if (!isok(n), return(-1)); my(x = sqrtint(n)); while (! issquare(n*(x^2 - n)), x++); x^2-n; \\ Michel Marcus, May 24 2025

cp's OEIS Frontend

A383734 Numbers k such that 2+k and 2*k are squares.

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