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A219394 Numbers k such that 17*k+1 is a square.

%I A219394 #32 Mar 15 2022 04:35:13
%S A219394 0,15,19,64,72,147,159,264,280,415,435,600,624,819,847,1072,1104,1359,
%T A219394 1395,1680,1720,2035,2079,2424,2472,2847,2899,3304,3360,3795,3855,
%U A219394 4320,4384,4879,4947,5472,5544,6099,6175,6760,6840,7455,7539,8184,8272,8947
%N A219394 Numbers k such that 17*k+1 is a square.
%C A219394 Equivalently, numbers of the form m*(17*m+2), where m = 0,-1,1,-2,2,-3,3,...
%C A219394 Also, integer values of h*(h+2)/17.
%H A219394 Bruno Berselli, <a href="/A219394/b219394.txt">Table of n, a(n) for n = 1..1000</a>
%H A219394 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F A219394 G.f.: x^2*(15+4*x+15*x^2)/((1+x)^2*(1-x)^3).
%F A219394 a(n) = a(-n+1) = (34*n*(n-1)+13*(-1)^n*(2*n-1)+5)/8 + 1.
%F A219394 Sum_{n>=2} 1/a(n) = 17/4 - cot(2*Pi/17)*Pi/2. - _Amiram Eldar_, Mar 15 2022
%p A219394 A219394:=proc(q)
%p A219394 local n;
%p A219394 for n from 1 to q do if type(sqrt(17*n+1), integer) then print(n);
%p A219394 fi; od; end:
%p A219394 A219394(1000); # _Paolo P. Lava_, Feb 19 2013
%t A219394 Select[Range[0, 9000], IntegerQ[Sqrt[17 # + 1]] &]
%t A219394 CoefficientList[Series[x (15 + 4 x + 15 x^2)/((1 + x)^2 (1 - x)^3), {x, 0, 50}], x] (* _Vincenzo Librandi_, Aug 18 2013 *)
%t A219394 LinearRecurrence[{1,2,-2,-1,1},{0,15,19,64,72},50] (* _Harvey P. Dale_, May 01 2017 *)
%o A219394 (Magma) [n: n in [0..9000] | IsSquare(17*n+1)];
%o A219394 (Magma) I:=[0,15,19,64,72]; [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 A219394 Cf. similar sequences listed in A219257.
%K A219394 nonn,easy
%O A219394 1,2
%A A219394 _Bruno Berselli_, Dec 03 2012