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A219389 Numbers k such that 13*k+1 is a square.

%I A219389 #39 Mar 15 2022 04:35:52
%S A219389 0,11,15,48,56,111,123,200,216,315,335,456,480,623,651,816,848,1035,
%T A219389 1071,1280,1320,1551,1595,1848,1896,2171,2223,2520,2576,2895,2955,
%U A219389 3296,3360,3723,3791,4176,4248,4655,4731,5160,5240,5691,5775,6248,6336,6831,6923
%N A219389 Numbers k such that 13*k+1 is a square.
%C A219389 Equivalently, numbers of the form m*(13*m+2), where m = 0,-1,1,-2,2,-3,3,...
%C A219389 Also, integer values of h*(h+2)/13.
%H A219389 Bruno Berselli, <a href="/A219389/b219389.txt">Table of n, a(n) for n = 1..1000</a>
%H A219389 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F A219389 G.f.: x^2*(11+4*x+11*x^2)/((1+x)^2*(1-x)^3).
%F A219389 a(n) = a(-n+1) = (26*n*(n-1)+9*(-1)^n*(2*n-1)+1)/8 +1.
%F A219389 Sum_{n>=2} 1/a(n) = 13/4 - cot(2*Pi/13)*Pi/2. - _Amiram Eldar_, Mar 15 2022
%p A219389 A219389:=proc(q)
%p A219389 local n;
%p A219389 for n from 1 to q do if type(sqrt(13*n+1), integer) then print(n);
%p A219389 fi; od; end:
%p A219389 A219389(1000); # _Paolo P. Lava_, Feb 19 2013
%t A219389 Select[Range[0, 7000], IntegerQ[Sqrt[13 # + 1]] &]
%t A219389 CoefficientList[Series[x (11 + 4 x + 11 x^2)/((1 + x)^2 (1 - x)^3), {x, 0, 50}], x] (* _Vincenzo Librandi_, Aug 18 2013 *)
%o A219389 (Magma) [n: n in [0..7000] | IsSquare(13*n+1)];
%o A219389 (Magma) I:=[0,11,15,48,56]; [n le 5 select I[n] else Self(n-1)+2*Self(n-2)-2*Self(n-3)-Self(n-4)+Self(n-5): n in [1..50]]; // _Vincenzo Librandi_, Aug 18 2013
%Y A219389 Cf. similar sequences listed in A219257.
%Y A219389 Cf. A175886 (square roots of 13*a(n)+1).
%K A219389 nonn,easy
%O A219389 1,2
%A A219389 _Bruno Berselli_, Nov 19 2012