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

0, 21, 25, 88, 96, 201, 213, 360, 376, 565, 585, 816, 840, 1113, 1141, 1456, 1488, 1845, 1881, 2280, 2320, 2761, 2805, 3288, 3336, 3861, 3913, 4480, 4536, 5145, 5205, 5856, 5920, 6613, 6681, 7416, 7488, 8265, 8341, 9160, 9240, 10101, 10185, 11088, 11176, 12121, 12213

Offset: 1

Author

Bruno Berselli, Nov 24 2012

Keywords

• nonn
• easy

Comments

Equivalently, numbers of the form m*(23*m+2), where m = 0,-1,1,-2,2,-3,3,...

Also, integer values of h*(h+2)/23.

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..13000] | IsSquare(23*n+1)];

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

Maple

A219393:=proc(q)
local n;
for n from 1 to q do if type(sqrt(23*n+1), integer) then print(n);
fi; od; end:
A219393(1000); # Paolo P. Lava, Feb 19 2013

Mathematica

Select[Range[0, 13000], IntegerQ[Sqrt[23 # + 1]] &]
CoefficientList[Series[x (21 + 4 x + 21 x^2)/((1 + x)^2 (1 - x)^3), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 18 2013 *)
LinearRecurrence[{1,2,-2,-1,1},{0,21,25,88,96},50] (* Harvey P. Dale, Jun 22 2025 *)

Formula

G.f.: x^2*(21 + 4*x + 21*x^2)/((1 + x)^2*(1 - x)^3).

a(n) = a(-n+1) = (46*n*(n-1) + 19*(-1)^n*(2*n - 1) + 3)/8 + 2.

Sum_{n>=2} 1/a(n) = 23/4 - cot(2*Pi/23)*Pi/2. - Amiram Eldar, Mar 16 2022