A047444 Numbers that are congruent to {0, 3, 5, 6} mod 8.

0, 3, 5, 6, 8, 11, 13, 14, 16, 19, 21, 22, 24, 27, 29, 30, 32, 35, 37, 38, 40, 43, 45, 46, 48, 51, 53, 54, 56, 59, 61, 62, 64, 67, 69, 70, 72, 75, 77, 78, 80, 83, 85, 86, 88, 91, 93, 94, 96, 99, 101, 102, 104, 107, 109, 110, 112, 115, 117, 118, 120, 123, 125

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Links

• Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).

Crossrefs

Cf. A047398, A047645.

Programs

Magma

[n : n in [0..150] | n mod 8 in [0, 3, 5, 6]]; // Wesley Ivan Hurt, May 26 2016

Maple

A047444:=n->(1+I)*(4*n-4*n*I+3*I-3-I^(-n)+I^(1+n))/4: seq(A047444(n), n=1..100); # Wesley Ivan Hurt, May 26 2016

Mathematica

Table[(1+I)*(4n-4*n*I+3*I-3-I^(-n)+I^(1+n))/4, {n, 80}] (* Wesley Ivan Hurt, May 26 2016 *)
LinearRecurrence[{2,-2,2,-1},{0,3,5,6},70] (* Harvey P. Dale, Aug 26 2019 *)

Formula

G.f.: x^2*(3-x+2*x^2) / ( (x^2+1)*(x-1)^2 ). - R. J. Mathar, Dec 07 2011

From Wesley Ivan Hurt, May 26 2016: (Start)

a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) for n>4.

a(n) = (1+i)*(4*n-4*n*i+3*i-3-i^(-n)+i^(1+n))/4 where i=sqrt(-1).

a(2k) = A047398(k), a(2k-1) = A047645(k). (End)

E.g.f.: (4 - sin(x) - cos(x) + (4*x - 3)*exp(x))/2. - Ilya Gutkovskiy, May 27 2016

Sum_{n>=2} (-1)^n/a(n) = 3*log(2)/8 - (3-2*sqrt(2))*Pi/16. - Amiram Eldar, Dec 21 2021