A047605 Numbers that are congruent to {0, 2, 3, 5} mod 8.

0, 2, 3, 5, 8, 10, 11, 13, 16, 18, 19, 21, 24, 26, 27, 29, 32, 34, 35, 37, 40, 42, 43, 45, 48, 50, 51, 53, 56, 58, 59, 61, 64, 66, 67, 69, 72, 74, 75, 77, 80, 82, 83, 85, 88, 90, 91, 93, 96, 98, 99, 101, 104, 106, 107, 109, 112, 114, 115, 117, 120, 122, 123

Offset

1,2

Links

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

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

Formula

From Bruno Berselli, Dec 05 2011: (Start)

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

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

From Wesley Ivan Hurt, Jun 04 2016: (Start)

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

a(2k) = A047617(k), a(2k-1) = A047470(k). (End)

E.g.f.: (6 + sin(x) - cos(x) + (4*x - 5)*exp(x))/2. - Ilya Gutkovskiy, Jun 05 2016

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

Maple

A047605:=n->2*(n-1)-(I^(n*(n+1))+1)/2: seq(A047605(n), n=1..100); # Wesley Ivan Hurt, Jun 04 2016

Mathematica

Table[(1+I)*((4-4*I)*n+5*I-5+I^(1-n)-I^n)/4, {n, 80}] (* Wesley Ivan Hurt, Jun 04 2016 *)
Flatten[#+{0, 2, 3, 5}&/@(8*Range[0, 20])] (* or *) LinearRecurrence[{2, -2, 2, -1}, {0, 2, 3, 5}, 100] (* Harvey P. Dale, Sep 30 2018 *)

Prog

(PARI) a(n)=n\4*8+[-3, 0, 2, 3][n%4+1] \\ Charles R Greathouse IV, Dec 05 2011
(Magma) [n : n in [0..150] | n mod 8 in [0, 2, 3, 5]]; // Wesley Ivan Hurt, Jun 04 2016

Crossrefs

Cf. A047470, A047617.

Sequence in context: A028841 A028840 A189143 * A295085 A153000 A222172

Adjacent sequences: A047602 A047603 A047604 * A047606 A047607 A047608

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved