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

1,1

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).

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

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 *)

Prog

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

Crossrefs

Cf. A062717, A104777 (the squares 6*k+7).

Sequence in context: A203321 A203319 A366171 * A171140 A186452 A016046

Adjacent sequences: A164094 A164095 A164096 * A164098 A164099 A164100

Keyword

nonn,easy

Author

Vincenzo Librandi, Aug 10 2009

Extensions

Edited by R. J. Mathar, Aug 26 2009

Status

approved