A047575 Numbers that are congruent to {0, 5, 6, 7} mod 8.

0, 5, 6, 7, 8, 13, 14, 15, 16, 21, 22, 23, 24, 29, 30, 31, 32, 37, 38, 39, 40, 45, 46, 47, 48, 53, 54, 55, 56, 61, 62, 63, 64, 69, 70, 71, 72, 77, 78, 79, 80, 85, 86, 87, 88, 93, 94, 95, 96, 101, 102, 103, 104, 109, 110, 111, 112, 117, 118, 119, 120, 125

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Links

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

Crossrefs

Cf. A047451, A047550.

Programs

Magma

[n : n in [0..150] | n mod 8 in [0, 5, 6, 7]]; // Wesley Ivan Hurt, May 29 2016

Maple

A047575:=n->(4*n-1+I^(2*n)-(1+I)*I^(-n)-(1-I)*I^n)/2: seq(A047575(n), n=1..100); # Wesley Ivan Hurt, May 29 2016

Mathematica

Select[Range[0,120], MemberQ[{0,5,6,7}, Mod[#,8]]&] (* Harvey P. Dale, Jun 30 2011 *)

Formula

From Wesley Ivan Hurt, May 29 2016: (Start)

G.f.: x^2*(5+x+x^2+x^3) / ((x-1)^2*(1+x+x^2+x^3)).

a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.

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

a(2k) = A047550(k), a(2k-1) = A047451(k). (End)

E.g.f.: 1 - sin(x) - cos(x) - sinh(x) + 2*x*exp(x). - Ilya Gutkovskiy, May 30 2016

Sum_{n>=2} (-1)^n/a(n) = 5*log(2)/8 - (2*sqrt(2)-1)*Pi/16. - Amiram Eldar, Dec 23 2021