A164097 Numbers k such that 6*k + 7 is a perfect square.

3, 7, 19, 27, 47, 59, 87, 103, 139, 159, 203, 227, 279, 307, 367, 399, 467, 503, 579, 619, 703, 747, 839, 887, 987, 1039, 1147, 1203, 1319, 1379, 1503, 1567, 1699, 1767, 1907, 1979, 2127, 2203, 2359, 2439, 2603, 2687, 2859, 2947, 3127, 3219, 3407, 3503, 3699

Offset: 1

Author

Vincenzo Librandi, Aug 10 2009

Keywords

• nonn
• easy

Comments

The entries are prime, or divisible by 3, or divisible by prime of the form 3*m+1.

Links

• Vincenzo Librandi, 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. A062717, A104777 (the squares 6*k+7).

Programs

Magma

[n: n in [1..4000] | IsSquare(6*n+7)]; // Vincenzo Librandi, Oct 12 2012

Mathematica

Select[Range[4000], IntegerQ[Sqrt[6 # + 7 ]] &] (* or *) LinearRecurrence[{1, 2, -2, -1, 1}, {3, 7, 19, 27, 47}, 50] (* Harvey P. Dale, Apr 29 2011 *)

Formula

From R. J. Mathar, Aug 26 2009: (Start)

a(n) = a(n-1) +2*a(n-2) -2*a(n-3) -a(n-4) +a(n-5).

G.f.: x*(-3-4*x-6*x^2+x^4)/((1+x)^2*(x-1)^3).

a(n) = 3*(2*n-1+2*n^2)/4 -(-1)^n*(1+2*n)/4 = A062717(n+1)-1. (End)

Sum_{n>=1} 1/a(n) = 1 + (tan((2+sqrt(7))*Pi/6) - cot((1+sqrt(7))*Pi/6))*Pi/(2*sqrt(7)). - Amiram Eldar, Feb 24 2023

Extensions

Edited by R. J. Mathar, Aug 26 2009