A047600 Numbers that are congruent to {1, 3, 4, 5} mod 8.

1, 3, 4, 5, 9, 11, 12, 13, 17, 19, 20, 21, 25, 27, 28, 29, 33, 35, 36, 37, 41, 43, 44, 45, 49, 51, 52, 53, 57, 59, 60, 61, 65, 67, 68, 69, 73, 75, 76, 77, 81, 83, 84, 85, 89, 91, 92, 93, 97, 99, 100, 101, 105, 107, 108, 109, 113, 115, 116, 117, 121, 123, 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+2*x+x^2+x^3+3*x^4)/((1+x)*(1-x)^2*(1+x^2)).

a(n) = 2*n-1 -(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) = A047621(k), a(2k-1) = A047461(k). (End)

E.g.f.: (6 + sin(x) - 2*cos(x) + (4*x - 3)*sinh(x) + 4*(x - 1)*cosh(x))/2. - Ilya Gutkovskiy, Jun 03 2016

Sum_{n>=1} (-1)^(n+1)/a(n) = (sqrt(2)+1)*Pi/16 - (4+3*sqrt(2))*log(2)/16 + 3*sqrt(2)*log(sqrt(2)+2)/8. - Amiram Eldar, Dec 24 2021

Maple

A047600:=n->2*n-1-(3+I^(2*n))*(1+I^(n*(n+1)))/4: seq(A047600(n), n=1..100); # Wesley Ivan Hurt, Jun 02 2016

Mathematica

Select[Range[120], MemberQ[{1, 3, 4, 5}, Mod[#, 8]]&]  (* Harvey P. Dale, Mar 09 2011 *)
LinearRecurrence[{1, 0, 0, 1, -1}, {1, 3, 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, 3, 4, 5]];
(Maxima) makelist(2*n-1-(3+(-1)^n)*(1+%i^(n*(n+1)))/4, n, 1, 60);
(PARI) Vec((1+2*x+x^2+x^3+3*x^4)/((1+x)*(1-x)^2*(1+x^2))+O(x^60)) (End)

Crossrefs

Cf. A047461, A047621.

Sequence in context: A217040 A295718 A035236 * A325464 A010384 A028954

Adjacent sequences: A047597 A047598 A047599 * A047601 A047602 A047603

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved