A219392 Numbers k such that 22*k+1 is a square.

0, 20, 24, 84, 92, 192, 204, 344, 360, 540, 560, 780, 804, 1064, 1092, 1392, 1424, 1764, 1800, 2180, 2220, 2640, 2684, 3144, 3192, 3692, 3744, 4284, 4340, 4920, 4980, 5600, 5664, 6324, 6392, 7092, 7164, 7904, 7980, 8760, 8840, 9660, 9744, 10604, 10692

Offset: 1

Author

Bruno Berselli, Nov 22 2012

Keywords

• nonn
• easy

Comments

Equivalently, numbers of the form m*(22*m+2), where m = 0,-1,1,-2,2,-3,3,...

Also, integer values of 2*h*(h+1)/11.

Links

• Bruno Berselli, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Crossrefs

Cf. similar sequences listed in A219257.

Programs

Magma

[n: n in [0..11000] | IsSquare(22*n+1)];

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

Maple

A219392:=proc(q)
local n;
for n from 1 to q do if type(sqrt(22*n+1), integer) then print(n);
fi; od; end:
A219392(1000); # Paolo P. Lava, Feb 19 2013

Mathematica

Select[Range[0, 11000], IntegerQ[Sqrt[22 # + 1]] &]
CoefficientList[Series[4 x (5 + x + 5 x^2)/((1 + x)^2 (1 - x)^3), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 18 2013 *)

Formula

G.f.: 4*x^2*(5 + x + 5*x^2)/((1 + x)^2*(1 - x)^3).

a(n) = a(-n+1) = (22*n*(n-1) + 9*(-1)^n*(2*n - 1) + 1)/4 + 2.

Sum_{n>=2} 1/a(n) = 11/2 - cot(Pi/11)*Pi/2. - Amiram Eldar, Mar 16 2022