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A247215 Integers k such that 3k+1 and 6k+1 are both squares.

%I A247215 #20 Mar 25 2025 11:17:23
%S A247215 0,8,280,9520,323408,10986360,373212840,12678250208,430687294240,
%T A247215 14630689753960,497012764340408,16883803297819920,573552299361536880,
%U A247215 19483894374994434008,661878856450449219400,22484397224940279025600,763807626791519037651008
%N A247215 Integers k such that 3k+1 and 6k+1 are both squares.
%H A247215 Colin Barker, <a href="/A247215/b247215.txt">Table of n, a(n) for n = 1..650</a>
%H A247215 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (35,-35,1).
%F A247215 a(n) = (1/72)*(3*(3*(17-12*sqrt(2))^n+2*sqrt(2)*(17-12*sqrt(2))^n+3*(17+12*sqrt(2))^n-2*sqrt(2)*(17+12*sqrt(2))^n)-18).
%F A247215 From _Colin Barker_, Nov 26 2014: (Start)
%F A247215 a(n) = 8*A029546(n).
%F A247215 a(n) = 35*a(n-1)-35*a(n-2)+a(n-3).
%F A247215 G.f.: -8*x^2 / ((x-1)*(x^2-34*x+1)).
%F A247215 (End)
%F A247215 Lim_{n -> infinity} a(n+1)/a(n) = 33.970562748... = (1+sqrt(2))^4 (the dominant root of x^2-34*x+1). - _Joerg Arndt_, Dec 01 2014
%e A247215 When n=1, a(1)=0, 3(0)+1=1, 6(0)+1=1.
%e A247215 When n=2, a(2)=8, 3(8)+1=25, 6(8)+1=49.
%e A247215 When n=3, a(3)=280, 3(280)+1=841=29^2, 6(280)+1=1681=41^2.
%e A247215 When n=4, a(4)=9520, 3(9520)+1=28560=169^2, 6(9520)+1=57121=239^2.
%t A247215 LinearRecurrence[{35,-35,1},{0,8,280},20] (* _Harvey P. Dale_, Mar 25 2025 *)
%o A247215 (PARI) concat(0, Vec(-8*x^2/((x-1)*(x^2-34*x+1)) + O(x^100))) \\ _Colin Barker_, Nov 26 2014
%Y A247215 The common terms of A062717 and A001082.
%K A247215 nonn,easy
%O A247215 1,2
%A A247215 _Casey Leung_, Nov 26 2014
%E A247215 More terms from _Colin Barker_, Nov 26 2014