A047613 Numbers that are congruent to {1, 2, 4, 5} mod 8.

1, 2, 4, 5, 9, 10, 12, 13, 17, 18, 20, 21, 25, 26, 28, 29, 33, 34, 36, 37, 41, 42, 44, 45, 49, 50, 52, 53, 57, 58, 60, 61, 65, 66, 68, 69, 73, 74, 76, 77, 81, 82, 84, 85, 89, 90, 92, 93, 97, 98, 100, 101, 105, 106, 108, 109, 113, 114, 116, 117, 121, 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 17 2012: (Start)

G.f.: x*(1+x+2*x^2+x^3+3*x^4)/((1+x)*(1-x)^2*(1+x^2)).

a(n) = 2*n-2-((-1)^n+i^(n*(n+1)))/2, 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) = A047461(k). (End)

E.g.f.: (6 + sin(x) - cos(x) + (4*x - 3)*sinh(x) + (4*x - 5)*cosh(x))/2. - Ilya Gutkovskiy, Jun 02 2016

Sum_{n>=1} (-1)^(n+1)/a(n) = (2*sqrt(2)-1)*Pi/16 + sqrt(2)*log(sqrt(2)+2)/4 - (sqrt(2)+1)*log(2)/8. - Amiram Eldar, Dec 23 2021

Maple

A047613:=n->2*n-2-(I^(2*n)+I^(n*(n+1)))/2: seq(A047613(n), n=1..100); # Wesley Ivan Hurt, Jun 02 2016

Mathematica

Select[Range[120], MemberQ[{1, 2, 4, 5}, Mod[#, 8]] &] (* or *) LinearRecurrence[{1, 0, 0, 1, -1}, {1, 2, 4, 5, 9}, 60] (* Bruno Berselli, Jul 17 2012 *)

Prog

From Bruno Berselli, Jul 17 2012: (Start)
(Magma) [n: n in [1..120] | n mod 8 in [1, 2, 4, 5]];
(Maxima) makelist(2*n-2-((-1)^n+%i^(n*(n+1)))/2, n, 1, 60);
(PARI) Vec((1+x+2*x^2+x^3+3*x^4)/((1+x)*(1-x)^2*(1+x^2))+O(x^60)) (End)

Crossrefs

Cf. A047461, A047617.

Sequence in context: A284389 A342735 A342750 * A036795 A307563 A289175

Adjacent sequences: A047610 A047611 A047612 * A047614 A047615 A047616

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved