A219259 Numbers k such that 25*k+1 is a square.

0, 23, 27, 96, 104, 219, 231, 392, 408, 615, 635, 888, 912, 1211, 1239, 1584, 1616, 2007, 2043, 2480, 2520, 3003, 3047, 3576, 3624, 4199, 4251, 4872, 4928, 5595, 5655, 6368, 6432, 7191, 7259, 8064, 8136, 8987, 9063, 9960, 10040, 10983, 11067, 12056, 12144

Offset: 1

Author

Bruno Berselli, Nov 19 2012

Keywords

• nonn
• easy

Comments

Equivalently, numbers of the form m*(25*m+2), where m = 0,-1,1,-2,2,-3,3,...

Also, integer values of h*(h+2)/25.

Exponents in the expansion of Product_{n >= 1} (1 - q^(50*n))*(1 - q^(50*n-23))*(1 - q^(50*n-27)) = 1 - q^23 - q^27 + q^96 + q^104 - q^219 - q^231 + + - - .... - Peter Bala, Dec 18 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. A047209, A197652.

Programs

Magma

[n: n in [0..13000] | IsSquare(25*n+1)];

Magma

I:=[0,23,27,96,104]; [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

A219259:=proc(q)
local n;
for n from 1 to q do if type(sqrt(25*n+1), integer) then print(n);
fi; od; end:
A219259(1000); # Paolo P. Lava, Feb 19 2013

Mathematica

Select[Range[0, 13000], IntegerQ[Sqrt[25 # + 1]] &]
CoefficientList[Series[x (23 + 4 x + 23 x^2)/((1 + x)^2 (1 - x)^3), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 18 2013 *)

Formula

G.f.: x^2*(23 + 4*x + 23*x^2)/((1 + x)^2*(1 - x)^3).

a(n) = a(-n+1) = (50*n*(n-1) + 21*(-1)^n*(2*n - 1) + 5)/8 + 2.

25*a(n)+1 = A047209(A197652(n+1))^2.

Sum_{n>=2} 1/a(n) = 25/4 - cot(2*Pi/25)*Pi/2. - Amiram Eldar, Mar 17 2022