A047441 Numbers that are congruent to {0, 2, 5, 6} mod 8.

0, 2, 5, 6, 8, 10, 13, 14, 16, 18, 21, 22, 24, 26, 29, 30, 32, 34, 37, 38, 40, 42, 45, 46, 48, 50, 53, 54, 56, 58, 61, 62, 64, 66, 69, 70, 72, 74, 77, 78, 80, 82, 85, 86, 88, 90, 93, 94, 96, 98, 101, 102, 104, 106, 109, 110, 112, 114, 117, 118, 120, 122, 125

Offset

1,2

Links

Table of n, a(n) for n=1..63.

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

Formula

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

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

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

a(n) = (8*n-7-i^(2*n)-i^(1-n)+i^(1+n))/4 where i=sqrt(-1).

a(2k) = A130824(k) k>0, a(2k-1) = A047615(k). (End)

E.g.f.: (4 - sin(x) + (4*x - 3)*sinh(x) + 4*(x - 1)*cosh(x))/2. - Ilya Gutkovskiy, May 27 2016

a(n) = (8*n-7-cos(n*Pi)-2*sin(n*Pi/2))/4. - Wesley Ivan Hurt, Oct 05 2017

Sum_{n>=2} (-1)^n/a(n) = (sqrt(2)-1)*Pi/16 + (4-sqrt(2))*log(2)/16 + sqrt(2)*log(2+sqrt(2))/8. - Amiram Eldar, Dec 21 2021

Maple

A047441:=n->(8*n-7-I^(2*n)-I^(1-n)+I^(1+n))/4: seq(A047441(n), n=1..100); # Wesley Ivan Hurt, May 26 2016

Mathematica

Table[(8n-7-I^(2n)-I^(1-n)+I^(1+n))/4, {n, 80}] (* Wesley Ivan Hurt, May 26 2016 *)
LinearRecurrence[{1, 0, 0, 1, -1}, {0, 2, 5, 6, 8}, 100] (* Harvey P. Dale, Dec 25 2023 *)

Prog

(Magma) [n : n in [0..150] | n mod 8 in [0, 2, 5, 6]]; // Wesley Ivan Hurt, May 26 2016

Crossrefs

Cf. A047615, A130824.

Sequence in context: A121411 A224889 A363676 * A284777 A344312 A081083

Adjacent sequences: A047438 A047439 A047440 * A047442 A047443 A047444

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved