A219395 Numbers k such that 18*k+1 is a square.

0, 16, 20, 68, 76, 156, 168, 280, 296, 440, 460, 636, 660, 868, 896, 1136, 1168, 1440, 1476, 1780, 1820, 2156, 2200, 2568, 2616, 3016, 3068, 3500, 3556, 4020, 4080, 4576, 4640, 5168, 5236, 5796, 5868, 6460, 6536, 7160, 7240, 7896, 7980, 8668, 8756, 9476, 9568

Offset: 1

Author

Bruno Berselli, Dec 03 2012

Keywords

• nonn
• easy

Comments

Equivalently, numbers of the form m*(18*m+2), where m = 0,-1,1,-2,2,-3,3,...

Also, integer values of 2*h*(h+1)/9.

The sequence terms are the exponents in the expansion of Product_{n >= 1} (1 - q^(36*n))*(1 - q^(36*n-16))*(1 - q^(36*n-20)) = 1 - q^16 - q^20 + q^68 + q^76 - q^156 - q^168 + + - - .... - Peter Bala, Dec 24 2024

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..10000] | IsSquare(18*n+1)];

Magma

I:=[0,16,20,68,76]; [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

A219395:=proc(q)
local n;
for n from 1 to q do if type(sqrt(18*n+1), integer) then print(n);
fi; od; end:
A219395(1000); # Paolo P. Lava, Feb 19 2013

Mathematica

Select[Range[0, 10000], IntegerQ[Sqrt[18 # + 1]] &]
CoefficientList[Series[4 x (4 + x + 4 x^2)/((1 + x)^2 (1 - x)^3), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 18 2013 *)
LinearRecurrence[{1,2,-2,-1,1},{0,16,20,68,76},50] (* Harvey P. Dale, Dec 24 2014 *)

Formula

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

a(n) = a(-n+1) = (18*n*(n-1) + 7*(-1)^n*(2*n-1) - 1)/4 + 2.

Sum_{n>=2} 1/a(n) = 9/2 - cot(Pi/9)*Pi/2. - Amiram Eldar, Mar 15 2022