A219394 Numbers k such that 17*k+1 is a square.

0, 15, 19, 64, 72, 147, 159, 264, 280, 415, 435, 600, 624, 819, 847, 1072, 1104, 1359, 1395, 1680, 1720, 2035, 2079, 2424, 2472, 2847, 2899, 3304, 3360, 3795, 3855, 4320, 4384, 4879, 4947, 5472, 5544, 6099, 6175, 6760, 6840, 7455, 7539, 8184, 8272, 8947

Offset: 1

Author

Bruno Berselli, Dec 03 2012

Keywords

• nonn
• easy

Comments

Equivalently, numbers of the form m*(17*m+2), where m = 0,-1,1,-2,2,-3,3,...

Also, integer values of h*(h+2)/17.

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..9000] | IsSquare(17*n+1)];

Magma

I:=[0,15,19,64,72]; [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

A219394:=proc(q)
local n;
for n from 1 to q do if type(sqrt(17*n+1), integer) then print(n);
fi; od; end:
A219394(1000); # Paolo P. Lava, Feb 19 2013

Mathematica

Select[Range[0, 9000], IntegerQ[Sqrt[17 # + 1]] &]
CoefficientList[Series[x (15 + 4 x + 15 x^2)/((1 + x)^2 (1 - x)^3), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 18 2013 *)
LinearRecurrence[{1,2,-2,-1,1},{0,15,19,64,72},50] (* Harvey P. Dale, May 01 2017 *)

Formula

G.f.: x^2*(15+4*x+15*x^2)/((1+x)^2*(1-x)^3).

a(n) = a(-n+1) = (34*n*(n-1)+13*(-1)^n*(2*n-1)+5)/8 + 1.

Sum_{n>=2} 1/a(n) = 17/4 - cot(2*Pi/17)*Pi/2. - Amiram Eldar, Mar 15 2022