A047606 Numbers that are congruent to {1, 2, 3, 5} mod 8.

1, 2, 3, 5, 9, 10, 11, 13, 17, 18, 19, 21, 25, 26, 27, 29, 33, 34, 35, 37, 41, 42, 43, 45, 49, 50, 51, 53, 57, 58, 59, 61, 65, 66, 67, 69, 73, 74, 75, 77, 81, 82, 83, 85, 89, 90, 91, 93, 97, 98, 99, 101, 105, 106, 107, 109, 113, 114, 115, 117, 121, 122, 123

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

a(n) = 2*n-3+(3-(-1)^n)*(1-i^(n*(n+1)))/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) = A047471(k). (End)

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

Sum_{n>=1} (-1)^(n+1)/a(n) = (3*sqrt(2)-2)*Pi/16 + (2-sqrt(2))*log(2)/16 + sqrt(2)*log(2+sqrt(2))/8. - Amiram Eldar, Dec 23 2021

Maple

A047606:=n->2*n-3+(3-I^(2*n))*(1-I^(n*(n+1)))/4: seq(A047606(n), n=1..100); # Wesley Ivan Hurt, Jun 02 2016

Mathematica

Select[Range[120], MemberQ[{1, 2, 3, 5}, Mod[#, 8]] &] (* or *) LinearRecurrence[{1, 0, 0, 1, -1}, {1, 2, 3, 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, 3, 5]];
(Maxima) makelist(2*n-3+(3-(-1)^n)*(1-%i^(n*(n+1)))/4, n, 1, 60);
(PARI) Vec((1+x+x^2+2*x^3+3*x^4)/((1+x)*(1-x)^2*(1+x^2))+O(x^60)) (End)

Crossrefs

Cf. A047471, A047617.

Sequence in context: A058314 A072735 A127149 * A226826 A333607 A047370

Adjacent sequences: A047603 A047604 A047605 * A047607 A047608 A047609

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved