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

%I A382209 #102 May 22 2025 05:18:59
%S A382209 90,136890,197402490,284654260890,410471246808090,591899253243012090,
%T A382209 853518312705176632890,1230772815021611461622490,
%U A382209 1774773545742851022483004890,2559222222188376152809031436090,3690396669622092669499600847844090,5321549438372835441042271613559748890
%N A382209 Numbers k such that 10+k and 10*k are perfect squares.
%C A382209 The limit of a(n+1)/a(n) is 1441.99930651839... = 721+228*sqrt(10) = (19+6*sqrt(10))^2.
%C A382209 If 10*A158490(n) is a perfect square, then A158490(n) is a term.
%H A382209 Emilio Martín, <a href="/A382209/b382209.txt">Table of n, a(n) for n = 1..100</a>
%H A382209 Wikipedia, <a href="https://de.wikipedia.org/wiki/Pellsche_Gleichung">Negative Pell equation</a> (in German)
%H A382209 Wikipedia, <a href="https://en.wikipedia.org/wiki/Pell%27s_equation">Pell's equation</a>
%H A382209 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1443,-1443,1).
%F A382209 a(n) = 10 * ((1/2) * (3+sqrt(10))^(2*n-1) + (1/2) * (3-sqrt(10))^(2*n-1))^2.
%F A382209 a(n) = 10 * (sinh((2n-1) * arcsinh(3)))^2.
%F A382209 a(n) = 10 * A173127(n)^2 = 100 * A097315(n)^2 - 10 (negative Pell's equation solutions).
%F A382209 a(n+2) = 1442 * a(n+1) - a(n) + 7200.
%F A382209 G.f.: 90*(1 + 78*x + x^2)/((1 - x)*(1 - 1442*x + x^2)). - _Stefano Spezia_, Apr 24 2025
%e A382209 90 is a term because 10+90=100 is a square and 10*90=900 is a square.
%e A382209 (3,1) is a solution to x^2 - 10*y^2 = -1, from which a(n) = 100*y^2-10 = 10*x^2 = 90.
%t A382209 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 *)
%o A382209 (Python)
%o A382209 from itertools import islice
%o A382209 def A382209_gen(): # generator of terms
%o A382209     x, y = 30, 10
%o A382209     while True:
%o A382209         yield x**2//10
%o A382209         x, y = x*19+y*60, x*6+y*19
%o A382209 A382209_list = list(islice(A382209_gen(),30)) # _Chai Wah Wu_, Apr 24 2025
%Y A382209 Subsequence of A158490.
%Y A382209 Cf. A097315, A005667, A173127, A081071.
%Y A382209 Cf. A383734 = 2*A008843 (2+k and 2*k are squares).
%Y A382209 Cf. 5*A075796^2 (5+k and 5*k are squares).
%Y A382209 Cf. 5*A081071 (20+k and 20*k are squares).
%Y A382209 Cf. A245226 (m such that k+m and k*m are squares).
%K A382209 nonn,easy
%O A382209 1,1
%A A382209 _Emilio Martín_, Mar 18 2025