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A219390 Numbers k such that 14*k+1 is a square.

%I A219390 #41 Mar 15 2022 04:35:37
%S A219390 0,12,16,52,60,120,132,216,232,340,360,492,516,672,700,880,912,1116,
%T A219390 1152,1380,1420,1672,1716,1992,2040,2340,2392,2716,2772,3120,3180,
%U A219390 3552,3616,4012,4080,4500,4572,5016,5092,5560,5640,6132,6216,6732,6820
%N A219390 Numbers k such that 14*k+1 is a square.
%C A219390 Equivalently, numbers of the form m*(14*m+2), where m = 0,-1,1,-2,2,-3,3,...
%C A219390 Also, integer values of 2*h*(h+1)/7.
%H A219390 Bruno Berselli, <a href="/A219390/b219390.txt">Table of n, a(n) for n = 1..1000</a>
%H A219390 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F A219390 G.f.: 4*x^2*(3+x+3*x^2)/((1+x)^2*(1-x)^3).
%F A219390 a(n) = a(-n+1) = (14*n*(n-1)+5*(-1)^n*(2*n-1)+1)/4 +1.
%F A219390 a(n) = 2*A219191(n).
%F A219390 Sum_{n>=2} 1/a(n) = 7/2 - cot(Pi/7)*Pi/2. - _Amiram Eldar_, Mar 15 2022
%p A219390 A219390:=proc(q)
%p A219390 local n;
%p A219390 for n from 1 to q do if type(sqrt(14*n+1), integer) then print(n);
%p A219390 fi; od; end:
%p A219390 A219390(1000); # _Paolo P. Lava_, Feb 19 2013
%t A219390 Select[Range[0, 7000], IntegerQ[Sqrt[14 # + 1]] &]
%t A219390 CoefficientList[Series[4 x (3 + x + 3 x^2) ((1 + x)^2 (1 - x)^3), {x, 0, 50}], x] (* _Vincenzo Librandi_, Aug 18 2013 *)
%t A219390 LinearRecurrence[{1,2,-2,-1,1},{0,12,16,52,60},50] (* _Harvey P. Dale_, Feb 05 2019 *)
%o A219390 (Magma) [n: n in [0..7000] | IsSquare(14*n+1)];
%o A219390 (Magma) I:=[0,12,16,52,60]; [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 A219390 Cf. similar sequences listed in A219257.
%Y A219390 Cf. A113801 (square roots of 14*a(n)+1).
%K A219390 nonn,easy
%O A219390 1,2
%A A219390 _Bruno Berselli_, Nov 19 2012