A383734 -id:A383734 - cp's OEIS frontend

A382209 Numbers k such that 10+k and 10*k are perfect squares.

90, 136890, 197402490, 284654260890, 410471246808090, 591899253243012090, 853518312705176632890, 1230772815021611461622490, 1774773545742851022483004890, 2559222222188376152809031436090, 3690396669622092669499600847844090, 5321549438372835441042271613559748890

Offset: 1

Emilio Martín, Mar 18 2025

The limit of a(n+1)/a(n) is 1441.99930651839... = 721+228*sqrt(10) = (19+6*sqrt(10))^2.

If 10*A158490(n) is a perfect square, then A158490(n) is a term.

			90 is a term because 10+90=100 is a square and 10*90=900 is a square.
(3,1) is a solution to x^2 - 10*y^2 = -1, from which a(n) = 100*y^2-10 = 10*x^2 = 90.

Emilio Martín, Table of n, a(n) for n = 1..100
Wikipedia, Negative Pell equation (in German)
Wikipedia, Pell's equation
Index entries for linear recurrences with constant coefficients, signature (1443,-1443,1).

Subsequence of A158490.
Cf. A097315, A005667, A173127, A081071.
Cf. A383734 = 2*A008843 (2+k and 2*k are squares).
Cf. 5*A075796^2 (5+k and 5*k are squares).
Cf. 5*A081071 (20+k and 20*k are squares).
Cf. A245226 (m such that k+m and k*m are squares).

CoefficientList[Series[ 90*(1 + 78*x + x^2)/((1 - x)*(1 - 1442*x + x^2)),{x,0,11}],x] (* or *) LinearRecurrence[{1443,-1443,1},{90,136890,197402490},12] (* James C. McMahon, May 08 2025 *)

from itertools import islice
def A382209_gen(): # generator of terms
    x, y = 30, 10
    while True:
        yield x**2//10
        x, y = x*19+y*60, x*6+y*19
A382209_list = list(islice(A382209_gen(),30)) # Chai Wah Wu, Apr 24 2025

a(n) = 10 * ((1/2) * (3+sqrt(10))^(2*n-1) + (1/2) * (3-sqrt(10))^(2*n-1))^2.
a(n) = 10 * (sinh((2n-1) * arcsinh(3)))^2.
a(n) = 10 * A173127(n)^2 = 100 * A097315(n)^2 - 10 (negative Pell's equation solutions).
a(n+2) = 1442 * a(n+1) - a(n) + 7200.
G.f.: 90*(1 + 78*x + x^2)/((1 - x)*(1 - 1442*x + x^2)). - Stefano Spezia, Apr 24 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

A382209 Numbers k such that 10+k and 10*k are perfect squares.

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