A047612 Numbers that are congruent to {0, 2, 4, 5} mod 8.

0, 2, 4, 5, 8, 10, 12, 13, 16, 18, 20, 21, 24, 26, 28, 29, 32, 34, 36, 37, 40, 42, 44, 45, 48, 50, 52, 53, 56, 58, 60, 61, 64, 66, 68, 69, 72, 74, 76, 77, 80, 82, 84, 85, 88, 90, 92, 93, 96, 98, 100, 101, 104, 106, 108, 109, 112, 114, 116, 117, 120, 122, 124

Offset

1,2

Links

Bruno Berselli, Table of n, a(n) for n = 1..1000

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

Formula

From Bruno Berselli, Jul 18 2012: (Start)

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

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

From Wesley Ivan Hurt, Jun 02 2016: (Start)

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

a(2k) = A047617(k), a(2k-1) = A008586(k-1) for k>0. (End)

E.g.f.: (6 - cos(x) + 4*(x - 1)*sinh(x) + (4*x - 5)*cosh(x))/2. - Ilya Gutkovskiy, Jun 03 2016

Sum_{n>=2} (-1)^n/a(n) = (2-sqrt(2))*Pi/16 + 5*log(2)/8 + sqrt(2)*log(sqrt(2)-1)/8. - Amiram Eldar, Dec 21 2021

Maple

A047612:=n->2*n-2-(1+I^(2*n))*(1+I^n)/4: seq(A047612(n), n=1..100); # Wesley Ivan Hurt, Jun 02 2016

Mathematica

Select[Range[0, 120], MemberQ[{0, 2, 4, 5}, Mod[#, 8]] &] (* or *) LinearRecurrence[{1, 0, 0, 1, -1}, {0, 2, 4, 5, 8}, 60] (* Bruno Berselli, Jul 18 2012 *)

Prog

From Bruno Berselli, Jul 18 2012: (Start)
(Magma) [n: n in [0..120] | n mod 8 in [0, 2, 4, 5]];
(Maxima) makelist(2*n-2-(1+(-1)^n)*(1+%i^n)/4, n, 1, 60);
(PARI) concat(0, Vec((2+2*x+x^2+3*x^3)/((1+x)*(1-x)^2*(1+x^2))+O(x^60))) (End)

Crossrefs

Cf. A008586, A047617.

Sequence in context: A046809 A112777 A188972 * A123886 A005242 A323976

Adjacent sequences: A047609 A047610 A047611 * A047613 A047614 A047615

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved