A219258 Numbers k such that 27*k+1 is a square.

0, 25, 29, 104, 112, 237, 249, 424, 440, 665, 685, 960, 984, 1309, 1337, 1712, 1744, 2169, 2205, 2680, 2720, 3245, 3289, 3864, 3912, 4537, 4589, 5264, 5320, 6045, 6105, 6880, 6944, 7769, 7837, 8712, 8784, 9709, 9785, 10760, 10840, 11865, 11949, 13024, 13112

Offset: 1

Author

Bruno Berselli, Nov 19 2012

Keywords

• nonn
• easy

Comments

Equivalently, numbers of the form m*(27*m+2), where m = 0,-1,1,-2,2,-3,3,...

Also, integer values of h*(h+2)/27.

The sequence terms are the exponents in the expansion of Product_{n >= 1} (1 - q^(54*n))*(1 - q^(54*n-25))*(1 - q^(54*n-29)) = 1 - q^25 - q^29 + q^104 + q^112 - q^237 - q^249 + + - - .... - 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.

Cf. A056081 (square roots of 27*a(n)+1).

Programs

Magma

[n: n in [0..14000] | IsSquare(27*n+1)];

Magma

I:=[0,25,29,104,112]; [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

A219258:=proc(q)
local n;
for n from 1 to q do if type(sqrt(27*n+1), integer) then print(n);
fi; od; end:
A219258(1000); # Paolo P. Lava, Feb 19 2013

Mathematica

Select[Range[0, 14000], IntegerQ[Sqrt[27 # + 1]] &]
CoefficientList[Series[x (25 + 4 x + 25 x^2)/((1 + x)^2 (1 - x)^3), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 18 2013 *)

Formula

G.f.: x^2*(25 + 4*x + 25*x^2)/((1 + x)^2*(1 - x)^3).

a(n) = a(-n+1) = (54*n*(n-1) + 23*(-1)^n*(2*n - 1) - 1)/8 + 3.

Sum_{n>=2} 1/a(n) = 27/4 - cot(2*Pi/27)*Pi/2. - Amiram Eldar, Mar 17 2022