A219389 Numbers k such that 13*k+1 is a square.

0, 11, 15, 48, 56, 111, 123, 200, 216, 315, 335, 456, 480, 623, 651, 816, 848, 1035, 1071, 1280, 1320, 1551, 1595, 1848, 1896, 2171, 2223, 2520, 2576, 2895, 2955, 3296, 3360, 3723, 3791, 4176, 4248, 4655, 4731, 5160, 5240, 5691, 5775, 6248, 6336, 6831, 6923

Offset: 1

Author

Bruno Berselli, Nov 19 2012

Keywords

• nonn
• easy

Comments

Equivalently, numbers of the form m*(13*m+2), where m = 0,-1,1,-2,2,-3,3,...

Also, integer values of h*(h+2)/13.

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.

Cf. A175886 (square roots of 13*a(n)+1).

Programs

Magma

[n: n in [0..7000] | IsSquare(13*n+1)];

Magma

I:=[0,11,15,48,56]; [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

A219389:=proc(q)
local n;
for n from 1 to q do if type(sqrt(13*n+1), integer) then print(n);
fi; od; end:
A219389(1000); # Paolo P. Lava, Feb 19 2013

Mathematica

Select[Range[0, 7000], IntegerQ[Sqrt[13 # + 1]] &]
CoefficientList[Series[x (11 + 4 x + 11 x^2)/((1 + x)^2 (1 - x)^3), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 18 2013 *)

Formula

G.f.: x^2*(11+4*x+11*x^2)/((1+x)^2*(1-x)^3).

a(n) = a(-n+1) = (26*n*(n-1)+9*(-1)^n*(2*n-1)+1)/8 +1.

Sum_{n>=2} 1/a(n) = 13/4 - cot(2*Pi/13)*Pi/2. - Amiram Eldar, Mar 15 2022