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

Author

Emilio Martín, Mar 18 2025

Keywords

• nonn
• easy

Comments

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.

Examples

			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.
		

Links

• 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).

Crossrefs

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).

Programs

Mathematica

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 *)

Python

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

Formula

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