A047412 Numbers that are congruent to {0, 1, 2, 4, 6} mod 8.

0, 1, 2, 4, 6, 8, 9, 10, 12, 14, 16, 17, 18, 20, 22, 24, 25, 26, 28, 30, 32, 33, 34, 36, 38, 40, 41, 42, 44, 46, 48, 49, 50, 52, 54, 56, 57, 58, 60, 62, 64, 65, 66, 68, 70, 72, 73, 74, 76, 78, 80, 81, 82, 84, 86, 88, 89, 90, 92, 94, 96, 97, 98, 100, 102, 104, 105

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Links

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

Programs

GAP

Filtered([0..105],n->n mod 8 = 0 or n mod 8 = 1 or n mod 8 = 2 or n mod 8 = 4 or n mod 8 = 6); # Muniru A Asiru, Oct 23 2018

Magma

[n : n in [0..140] | n mod 8 in [0, 1, 2, 4, 6] ]; // Vincenzo Librandi, Mar 01 2016

Maple

A047412:=n->8*floor(n/5)+[(0, 1, 2, 4, 6)][(n mod 5)+1]: seq(A047412(n), n=0..100); # Wesley Ivan Hurt, Aug 08 2016

Mathematica

Flatten[Table[8*n + {0, 1, 2, 4, 6}, {n, 0, 11}]] (* Alonso del Arte, Sep 21 2011 *)
Select[Range[0, 150], MemberQ[{0, 1, 2, 4, 6}, Mod[#, 8]] &] (* Vincenzo Librandi, Mar 01 2016 *)
LinearRecurrence[{1,0,0,0,1,-1},{0,1,2,4,6,8},100] (* Harvey P. Dale, Aug 06 2018 *)

PARI

a(n)=n\5*8 + [0, 1, 2, 4, 6][n%5+1] \\ Charles R Greathouse IV, Oct 27 2015

Formula

G.f.: x^2*(1+x+2*x^2+2*x^3+2*x^4) / ( (x^4+x^3+x^2+x+1)*(x-1)^2 ). - R. J. Mathar, Dec 05 2011

From Wesley Ivan Hurt, Aug 08 2016: (Start)

a(n) = a(n-1) + a(n-5) - a(n-6) for n > 6.

a(n) = a(n-5) + 8 for n > 5.

a(n) = (40*n - 55 - 2*(n mod 5) - 2*((n+1) mod 5) + 3*((n+2) mod 5) + 3*((n+3) mod 5) - 2*((n+4) mod 5))/25.

a(5*k) = 8*k-2, a(5*k-1) = 8*k-4, a(5*k-2) = 8*k-6, a(5*k-3) = 8*k-7, a(5*k-4) = 8*k-8. (End)

a(n) = (40*n-55+6*cos(2*Pi*(n-1)/5)+2*cos(2*Pi*n/5)+2*cos(4*Pi*n/5)-2*cos(2*Pi*(n+1)/5)-6*cos(Pi*(4*n+1)/5)+6*sin(Pi*(4*n+3)/10)+2*sin(Pi*(8*n+3)/10)-6*sin(Pi*(8*n+1)/10))/25. - Wesley Ivan Hurt, Oct 10 2018