A047553 Numbers that are congruent to {0, 2, 6, 7} mod 8.

0, 2, 6, 7, 8, 10, 14, 15, 16, 18, 22, 23, 24, 26, 30, 31, 32, 34, 38, 39, 40, 42, 46, 47, 48, 50, 54, 55, 56, 58, 62, 63, 64, 66, 70, 71, 72, 74, 78, 79, 80, 82, 86, 87, 88, 90, 94, 95, 96, 98, 102, 103, 104, 106, 110, 111, 112, 114, 118, 119, 120, 122, 126

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, A047524.

Programs

Magma

[n : n in [0..150] | n mod 8 in [0, 2, 6, 7]]; // Wesley Ivan Hurt, May 29 2016

Maple

A047553:=n->(8*n-5-I^(2*n)+(1-2*I)*I^(-n)+(1+2*I)*I^n)/4: seq(A047553(n), n=1..100); # Wesley Ivan Hurt, May 29 2016

Mathematica

Select[Range[0,200], MemberQ[{0,2,6,7}, Mod[#,8]]&] (* Harvey P. Dale, Aug 09 2013 *)

Formula

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

G.f.: x^2*(2+4*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) = (8*n-5-i^(2*n)+(1-2*i)*i^(-n)+(1+2*i)*i^n)/4 where i=sqrt(-1).

a(2k) = A047524(k), a(2k-1) = A047451(k). (End)

E.g.f.: (2 - 2*sin(x) + cos(x) + (4*x - 2)*sinh(x) + (4*x - 3)*cosh(x))/2. - Ilya Gutkovskiy, May 29 2016

Sum_{n>=2} (-1)^n/a(n) = (8-sqrt(2))*log(2)/16 + sqrt(2)*log(2+sqrt(2))/8 - (sqrt(2)-1)*Pi/16. - Amiram Eldar, Dec 21 2021