A219396 Numbers k such that 19*k+1 is a square.

0, 17, 21, 72, 80, 165, 177, 296, 312, 465, 485, 672, 696, 917, 945, 1200, 1232, 1521, 1557, 1880, 1920, 2277, 2321, 2712, 2760, 3185, 3237, 3696, 3752, 4245, 4305, 4832, 4896, 5457, 5525, 6120, 6192, 6821, 6897, 7560, 7640, 8337, 8421, 9152, 9240

Offset: 1

Author

Bruno Berselli, Dec 03 2012

Keywords

• nonn
• easy

Comments

Equivalently, numbers of the form m*(19*m+2), where m = 0, -1, 1, -2, 2, -3, 3,...

Also, integer values of h*(h+2)/19.

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(19*n+1)];

Magma

I:=[0, 17, 21, 72, 80]; [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

A219396:=proc(q)
local n;
for n from 1 to q do if type(sqrt(19*n+1), integer) then print(n);
fi; od; end:
A219396(1000); # Paolo P. Lava, Feb 19 2013

Mathematica

Select[Range[0, 10000], IntegerQ[Sqrt[19 # + 1]] &]
CoefficientList[Series[x (17 + 4 x + 17 x^2)/((1 + x)^2 (1 - x)^3), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 18 2013 *)
LinearRecurrence[{1,2,-2,-1,1},{0,17,21,72,80},60] (* Harvey P. Dale, Sep 08 2021 *)

Formula

G.f.: x^2*(17 + 4*x + 17*x^2)/((1 + x)^2*(1 - x)^3).

a(n) = a(-n+1) = (38*n*(n-1) + 15*(-1)^n*(2*n - 1)-1)/8 + 2.

Sum_{n>=2} 1/a(n) = 19/4 - cot(2*Pi/19)*Pi/2. - Amiram Eldar, Mar 15 2022

Extensions

Typo corrected in the first comment by Mokhtar Mohamed, Dec 03 2012