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%I #34 Mar 16 2022 05:49:37
%S 0,20,24,84,92,192,204,344,360,540,560,780,804,1064,1092,1392,1424,
%T 1764,1800,2180,2220,2640,2684,3144,3192,3692,3744,4284,4340,4920,
%U 4980,5600,5664,6324,6392,7092,7164,7904,7980,8760,8840,9660,9744,10604,10692
%N Numbers k such that 22*k+1 is a square.
%C Equivalently, numbers of the form m*(22*m+2), where m = 0,-1,1,-2,2,-3,3,...
%C Also, integer values of 2*h*(h+1)/11.
%H Bruno Berselli, <a href="/A219392/b219392.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F G.f.: 4*x^2*(5 + x + 5*x^2)/((1 + x)^2*(1 - x)^3).
%F a(n) = a(-n+1) = (22*n*(n-1) + 9*(-1)^n*(2*n - 1) + 1)/4 + 2.
%F Sum_{n>=2} 1/a(n) = 11/2 - cot(Pi/11)*Pi/2. - _Amiram Eldar_, Mar 16 2022
%p A219392:=proc(q)
%p local n;
%p for n from 1 to q do if type(sqrt(22*n+1), integer) then print(n);
%p fi; od; end:
%p A219392(1000); # _Paolo P. Lava_, Feb 19 2013
%t Select[Range[0, 11000], IntegerQ[Sqrt[22 # + 1]] &]
%t CoefficientList[Series[4 x (5 + x + 5 x^2)/((1 + x)^2 (1 - x)^3), {x, 0, 50}], x] (* _Vincenzo Librandi_, Aug 18 2013 *)
%o (Magma) [n: n in [0..11000] | IsSquare(22*n+1)];
%o (Magma) I:=[0,20,24,84,92]; [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 Cf. similar sequences listed in A219257.
%K nonn,easy
%O 1,2
%A _Bruno Berselli_, Nov 22 2012
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