A219390 Numbers k such that 14*k+1 is a square.

0, 12, 16, 52, 60, 120, 132, 216, 232, 340, 360, 492, 516, 672, 700, 880, 912, 1116, 1152, 1380, 1420, 1672, 1716, 1992, 2040, 2340, 2392, 2716, 2772, 3120, 3180, 3552, 3616, 4012, 4080, 4500, 4572, 5016, 5092, 5560, 5640, 6132, 6216, 6732, 6820

Offset: 1

Author

Bruno Berselli, Nov 19 2012

Keywords

• nonn
• easy

Comments

Equivalently, numbers of the form m*(14*m+2), where m = 0,-1,1,-2,2,-3,3,...

Also, integer values of 2*h*(h+1)/7.

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. A113801 (square roots of 14*a(n)+1).

Programs

Magma

[n: n in [0..7000] | IsSquare(14*n+1)];

Magma

I:=[0,12,16,52,60]; [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

A219390:=proc(q)
local n;
for n from 1 to q do if type(sqrt(14*n+1), integer) then print(n);
fi; od; end:
A219390(1000); # Paolo P. Lava, Feb 19 2013

Mathematica

Select[Range[0, 7000], IntegerQ[Sqrt[14 # + 1]] &]
CoefficientList[Series[4 x (3 + x + 3 x^2) ((1 + x)^2 (1 - x)^3), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 18 2013 *)
LinearRecurrence[{1,2,-2,-1,1},{0,12,16,52,60},50] (* Harvey P. Dale, Feb 05 2019 *)

Formula

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

a(n) = a(-n+1) = (14*n*(n-1)+5*(-1)^n*(2*n-1)+1)/4 +1.

a(n) = 2*A219191(n).

Sum_{n>=2} 1/a(n) = 7/2 - cot(Pi/7)*Pi/2. - Amiram Eldar, Mar 15 2022