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A219396 Numbers k such that 19*k+1 is a square.

%I A219396 #39 Mar 15 2022 04:34:42
%S A219396 0,17,21,72,80,165,177,296,312,465,485,672,696,917,945,1200,1232,1521,
%T A219396 1557,1880,1920,2277,2321,2712,2760,3185,3237,3696,3752,4245,4305,
%U A219396 4832,4896,5457,5525,6120,6192,6821,6897,7560,7640,8337,8421,9152,9240
%N A219396 Numbers k such that 19*k+1 is a square.
%C A219396 Equivalently, numbers of the form m*(19*m+2), where m = 0, -1, 1, -2, 2, -3, 3,...
%C A219396 Also, integer values of h*(h+2)/19.
%H A219396 Bruno Berselli, <a href="/A219396/b219396.txt">Table of n, a(n) for n = 1..1000</a>
%H A219396 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F A219396 G.f.: x^2*(17 + 4*x + 17*x^2)/((1 + x)^2*(1 - x)^3).
%F A219396 a(n) = a(-n+1) = (38*n*(n-1) + 15*(-1)^n*(2*n - 1)-1)/8 + 2.
%F A219396 Sum_{n>=2} 1/a(n) = 19/4 - cot(2*Pi/19)*Pi/2. - _Amiram Eldar_, Mar 15 2022
%p A219396 A219396:=proc(q)
%p A219396 local n;
%p A219396 for n from 1 to q do if type(sqrt(19*n+1), integer) then print(n);
%p A219396 fi; od; end:
%p A219396 A219396(1000); # _Paolo P. Lava_, Feb 19 2013
%t A219396 Select[Range[0, 10000], IntegerQ[Sqrt[19 # + 1]] &]
%t A219396 CoefficientList[Series[x (17 + 4 x + 17 x^2)/((1 + x)^2 (1 - x)^3), {x, 0, 50}], x] (* _Vincenzo Librandi_, Aug 18 2013 *)
%t A219396 LinearRecurrence[{1,2,-2,-1,1},{0,17,21,72,80},60] (* _Harvey P. Dale_, Sep 08 2021 *)
%o A219396 (Magma) [n: n in [0..10000] | IsSquare(19*n+1)];
%o A219396 (Magma) I:=[0, 17, 21, 72, 80]; [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 A219396 Cf. similar sequences listed in A219257.
%K A219396 nonn,easy
%O A219396 1,2
%A A219396 _Bruno Berselli_, Dec 03 2012
%E A219396 Typo corrected in the first comment by _Mokhtar Mohamed_, Dec 03 2012