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A219393 Numbers k such that 23*k+1 is a square.

%I A219393 #33 Jun 22 2025 12:29:41
%S A219393 0,21,25,88,96,201,213,360,376,565,585,816,840,1113,1141,1456,1488,
%T A219393 1845,1881,2280,2320,2761,2805,3288,3336,3861,3913,4480,4536,5145,
%U A219393 5205,5856,5920,6613,6681,7416,7488,8265,8341,9160,9240,10101,10185,11088,11176,12121,12213
%N A219393 Numbers k such that 23*k+1 is a square.
%C A219393 Equivalently, numbers of the form m*(23*m+2), where m = 0,-1,1,-2,2,-3,3,...
%C A219393 Also, integer values of h*(h+2)/23.
%H A219393 Bruno Berselli, <a href="/A219393/b219393.txt">Table of n, a(n) for n = 1..1000</a>
%H A219393 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F A219393 G.f.: x^2*(21 + 4*x + 21*x^2)/((1 + x)^2*(1 - x)^3).
%F A219393 a(n) = a(-n+1) = (46*n*(n-1) + 19*(-1)^n*(2*n - 1) + 3)/8 + 2.
%F A219393 Sum_{n>=2} 1/a(n) = 23/4 - cot(2*Pi/23)*Pi/2. - _Amiram Eldar_, Mar 16 2022
%p A219393 A219393:=proc(q)
%p A219393 local n;
%p A219393 for n from 1 to q do if type(sqrt(23*n+1), integer) then print(n);
%p A219393 fi; od; end:
%p A219393 A219393(1000); # _Paolo P. Lava_, Feb 19 2013
%t A219393 Select[Range[0, 13000], IntegerQ[Sqrt[23 # + 1]] &]
%t A219393 CoefficientList[Series[x (21 + 4 x + 21 x^2)/((1 + x)^2 (1 - x)^3), {x, 0, 50}], x] (* _Vincenzo Librandi_, Aug 18 2013 *)
%t A219393 LinearRecurrence[{1,2,-2,-1,1},{0,21,25,88,96},50] (* _Harvey P. Dale_, Jun 22 2025 *)
%o A219393 (Magma) [n: n in [0..13000] | IsSquare(23*n+1)];
%o A219393 (Magma) I:=[0,21,25,88,96]; [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 A219393 Cf. similar sequences listed in A219257.
%K A219393 nonn,easy
%O A219393 1,2
%A A219393 _Bruno Berselli_, Nov 24 2012