A047527 Numbers that are congruent to {0, 1, 2, 7} mod 8.

0, 1, 2, 7, 8, 9, 10, 15, 16, 17, 18, 23, 24, 25, 26, 31, 32, 33, 34, 39, 40, 41, 42, 47, 48, 49, 50, 55, 56, 57, 58, 63, 64, 65, 66, 71, 72, 73, 74, 79, 80, 81, 82, 87, 88, 89, 90, 95, 96, 97, 98, 103, 104, 105, 106, 111, 112, 113, 114, 119, 120

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Complement of numbers that are congruent to {3, 4, 5, 6} mod 8 (A047425). - Jaroslav Krizek, Dec 19 2009

Links

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

Crossrefs

Cf. A103127, A047425, A047467, A047522.

Programs

Magma

[n : n in [0..100] | n mod 8 in [0, 1, 2, 7]]; // Wesley Ivan Hurt, May 21 2016

Maple

seq(3*n-4*floor((n-2)/4)-6+(-1)^n, n=1..61); # Gary Detlefs, Mar 27 2010

Mathematica

Select[Range[0,200], MemberQ[{0,1,2,7}, Mod[#,8]]&] (* or *) LinearRecurrence[{1,0,0,1,-1}, {0,1,2,7,8}, 200] (* Harvey P. Dale, Sep 05 2014 *)

Formula

a(n) = 3*n-4*floor((n-2)/4)-6+(-1)^n. - Gary Detlefs, Mar 27 2010

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

a(n) = a(n-1) + a(n-4) - a(n-5) for n>5. - Harvey P. Dale, Sep 05 2014

From Wesley Ivan Hurt, May 21 2016: (Start)

a(n) = (4n-5+i^(2n)+(1+i)*i^(-n)+(1-i)*i^n)/2 where i = sqrt(-1).

a(2n) = A047522(n), a(2n-1) = A047467(n). (End)

Sum_{n>=2} (-1)^n/a(n) = (5-sqrt(2))*log(2)/8 + sqrt(2)*log(2+sqrt(2))/4 - Pi/16. - Amiram Eldar, Dec 20 2021