A047275 Numbers that are congruent to {0, 1, 6} mod 7.

0, 1, 6, 7, 8, 13, 14, 15, 20, 21, 22, 27, 28, 29, 34, 35, 36, 41, 42, 43, 48, 49, 50, 55, 56, 57, 62, 63, 64, 69, 70, 71, 76, 77, 78, 83, 84, 85, 90, 91, 92, 97, 98, 99, 104, 105, 106, 111, 112, 113, 118, 119, 120, 125, 126, 127, 132, 133, 134, 139, 140

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Nonnegative m such that floor(k*m^2/7) = k*floor(m^2/7), where k = 4, 5 or 6. See also the comment in A047299. [Bruno Berselli, Dec 03 2015]

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Crossrefs

Cf. A047299.

Programs

Magma

[n : n in [0..150] | n mod 7 in [0, 1, 6]]; // Wesley Ivan Hurt, Jun 10 2016

Maple

A047275:=n->(21*n-21+12*cos(2*n*Pi/3)+4*sqrt(3)*sin(2*n*Pi/3))/9: seq(A047275(n), n=1..100); # Wesley Ivan Hurt, Jun 10 2016

Mathematica

Select[Range[0, 120], Function[k, Mod[#, 7] == k] /@ Or[0, 1, 6] &] (* or *) Select[Range[0, 120], Function[k, Floor[k (#^2/7)] == k Floor[#^2/7]] /@ Or[4, 5, 6] &] (* Michael De Vlieger, Dec 03 2015 *)
LinearRecurrence[{1, 0, 1, -1}, {0, 1, 6, 7}, 100] (* Vincenzo Librandi, Jun 14 2016 *)

PARI

concat(0, Vec(x^2*(1+5*x+x^2)/((1+x+x^2)*(x-1)^2) + O(x^100))) \\ Altug Alkan, Dec 03 2015

Formula

G.f.: x^2*(1+5*x+x^2) / ((1+x+x^2)*(x-1)^2). - R. J. Mathar, Oct 25 2011

From Wesley Ivan Hurt, Jun 10 2016: (Start)

a(n) = a(n-1) + a(n-3) - a(n-4) for n>4.

a(n) = (21*n-21+12*cos(2*n*Pi/3)+4*sqrt(3)*sin(2*n*Pi/3))/9.

a(3k) = 7k-1, a(3k-1) = 7k-6, a(3k-2) = 7k-7. (End)