A032775 Numbers that are congruent to {0, 1, 2, 3, 5, 6} mod 7.

0, 1, 2, 3, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 47, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 68, 69, 70, 71, 72, 73, 75, 76, 77, 78, 79, 80, 82, 83

Offset: 1

Author

Patrick De Geest, May 15 1998

Keywords

• nonn
• easy

Comments

n(n+1)(n+2)...(n+6) / (n + (n+1) + (n+2) + ... + (n+6)) is an integer.

Links

• Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,1,-1).

Crossrefs

Cf. A032774, A032776.

Programs

Magma

[ n: n in [0..90] | n mod 7 in {0, 1, 2, 3, 5, 6} ]; // Vincenzo Librandi, Dec 29 2010

Maple

A032775:=n->(42*n-45-3*cos(n*Pi)+12*cos(n*Pi/3)-4*sqrt(3)*sin(2*n*Pi/3))/36: seq(A032775(n), n=1..100); # Wesley Ivan Hurt, Jun 15 2016

Mathematica

Select[Range[0, 100], MemberQ[{0, 1, 2, 3, 5, 6}, Mod[#, 7]] &] (* Wesley Ivan Hurt, Jun 15 2016 *)
DeleteCases[Range[0,100],?(Mod[#,7]==4&)] (* or *) LinearRecurrence[ {1,0,0,0,0,1,-1},{0,1,2,3,5,6,7},80] (* _Harvey P. Dale, Sep 19 2020 *)

Formula

Natural numbers minus '4, 11, 18, 25, ...' (= previous term + 7).

G.f.: x^2*(1+x+x^2+2*x^3+x^4+x^5) / ( (1+x)*(1+x+x^2)*(x^2-x+1)*(x-1)^2 ). - R. J. Mathar, Oct 08 2011

From Wesley Ivan Hurt, Jun 15 2016: (Start)

a(n) = a(n-1) + a(n-6) - a(n-7) for n > 7.

a(n) = (42*n - 45 - 3*cos(n*Pi) + 12*cos(n*Pi/3) - 4*sqrt(3)*sin(2*n*Pi/3))/36.

a(6k) = 7k-1, a(6k-1) = 7k-2, a(6k-2) = 7k-4, a(6k-3) = 7k-5, a(6k-4) = 7k-6, a(6k-5) = 7k-7. (End)