This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A343034 #25 Apr 04 2021 00:56:59
%S A343034 1,13,19,487,721,18493,27379,702247,1039681,26666893,39480499,
%T A343034 1012639687,1499219281,38453641213,56930852179,1460225726407,
%U A343034 2161873163521,55450123962253,82094249361619,2105644484839207,3117419602578001,79959040299927613,118379850648602419,3036337886912410087
%N A343034 Positive numbers m such that m^2 with last digit z deleted is still a perfect square k^2, and z divides m-k.
%C A343034 This sequence is the answer to the problem A1872 proposed on French mathematical site Diophante (see link).
%C A343034 Equivalent to the two Diophantine equations: m^2 = 10*k^2 + z and m-k = q*z for some q >= 1.
%C A343034 There exist solutions iff z = 1 or z = 9.
%C A343034 When (m, k, z) is a solution, then (19m+60k, 6m+19k, z) is another solution.
%C A343034 There is only one solution such that z = m-k: (13, 4, 9), see 1st example.
%C A343034 There exist two distinct families of solutions corresponding to z = 1 and z = 9, odd indices correspond to z = 1 and even indices to z = 9.
%C A343034 -> For z = 1, all solutions m of Pell equation m^2 - 10*k^2 = 1 are terms because z = 1 divides every m-k.
%C A343034 First few solutions (m, k) are (1, 0), (19, 6), (721, 228), (27379, 86568), ... with m = A078986(q) and corresponding k = 6*A078987(q).
%C A343034 -> For z = 9, solutions m must satisfy m^2 - 10*k^2 = 9 with 9 divides m-k. Among the 3 fundamental solutions (3, 0), (7, 2), (13, 4) of Pell equation m^2 -10*k^2 = 9, only (13, 4) gives solutions where 9 divides m-k.
%C A343034 First few solutions (m, k) are (13, 4), (487, 154), (18493, 5848), ... with m = A228209(3q).
%H A343034 Diophante, <a href="http://www.diophante.fr/problemes-par-themes/arithmetique-et-algebre/a1-pot-pourri/3697-a1872-carrement-magiques">A1872, Carrément magiques</a> (in French).
%H A343034 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,38,0,-1).
%F A343034 a(2n+1) = A078986(n) for n >= 0.
%F A343034 a(2n) = A228209(3n) for n >= 1.
%F A343034 a(n+4) = 38*a(n+2) - a(n), a(1) = 1, a(2) = 13, a(3)= 19, a(4) = 487.
%F A343034 G.f.: x*(1 + 13*x - 19*x^2 - 7*x^3)/(1 - 38*x^2 + x^4). - _Stefano Spezia_, Apr 03 2021
%e A343034 For m = 13, 13^2 = 169, 4^2 = 16, 13^2 - 10*4^2 = 9 and 9 = 13-4 divides 13-4.
%e A343034 For m = 19, 19^2 = 361, 6^2 = 36, 19^2 - 10*6^2 = 1 and 1 divides 19-6 = 13.
%e A343034 For m = 487, 487^2 = 237169, 154^2 = 23716, 487^2 - 10*154^2 = 9 and 9 divides 487-154 = 333 = 9*37.
%t A343034 LinearRecurrence[{0, 38, 0, -1}, {1, 13, 19, 487}, 24] (* _Amiram Eldar_, Apr 03 2021 *)
%Y A343034 Subsequence of A031149.
%Y A343034 A078986 is a subsequence.
%Y A343034 Cf. A078987, A228209.
%K A343034 nonn,easy,base
%O A343034 1,2
%A A343034 _Bernard Schott_, Apr 03 2021