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A059989 Numbers n such that 3*n+1 and 4*n+1 are both squares.

%I A059989 #36 Mar 04 2016 02:35:06
%S A059989 0,56,10920,2118480,410974256,79726887240,15466605150360,
%T A059989 3000441672282656,582070217817684960,112918621814958599640,
%U A059989 21905630561884150645256,4249579410383710266580080,824396499983877907565890320,159928671417461930357516142056
%N A059989 Numbers n such that 3*n+1 and 4*n+1 are both squares.
%H A059989 Colin Barker, <a href="/A059989/b059989.txt">Table of n, a(n) for n = 1..400</a>
%H A059989 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (195,-195,1).
%F A059989 a(n) = (A001570(n)^2 - 1)/3.
%F A059989 G.f.: 56*x^2 / (1-195*x+195*x^2-x^3).
%F A059989 From _Colin Barker_, Mar 03 2016: (Start)
%F A059989 a(n) = 195*a(n-1)-195*a(n-2)+a(n-3) for n>3.
%F A059989 a(n) = (-1)*((97+56*sqrt(3))^(-n)*(-1+(97+56*sqrt(3))^n)*(7+4*sqrt(3)+(-7+4*sqrt(3))*(97+56*sqrt(3))^n))/48.
%F A059989 (End)
%e A059989 3*56+1=13^2 and 4*56+1=15^2.
%p A059989 f:= proc(n) local u;
%p A059989   u:= <<7,8>|<6,7>>^n . <1,-1>;
%p A059989   (u[1]^2-1)/3
%p A059989 end proc:
%p A059989 map(f, [$1..30]); # _Robert Israel_, Mar 03 2016
%t A059989 CoefficientList[Series[56 x/(1 - 195 x + 195 x^2 - x^3), {x, 0, 13}], x] (* _Michael De Vlieger_, Mar 03 2016 *)
%o A059989 (PARI) isok(n) = issquare(3*n+1) && issquare(4*n+1) \\ _Michel Marcus_, Jun 08 2013
%o A059989 (PARI) concat(0, Vec(56*x^2/((1-x)*(1-194*x+x^2)) + O(x^20))) \\ _Colin Barker_, Mar 03 2016
%Y A059989 Cf. A245031.
%K A059989 nonn,easy
%O A059989 1,2
%A A059989 _David Radcliffe_, Mar 07 2001
%E A059989 Offset changed to 1 by _Joerg Arndt_, Mar 03 2016