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

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A047461, A047621.

Programs

Magma

[n: n in [1..120] | n mod 8 in [1,3,4,5]];

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 *)

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)

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