A047250 Numbers that are congruent to {0, 3, 4, 5} (mod 6).

0, 3, 4, 5, 6, 9, 10, 11, 12, 15, 16, 17, 18, 21, 22, 23, 24, 27, 28, 29, 30, 33, 34, 35, 36, 39, 40, 41, 42, 45, 46, 47, 48, 51, 52, 53, 54, 57, 58, 59, 60, 63, 64, 65, 66, 69, 70, 71, 72, 75, 76, 77, 78, 81, 82, 83, 84, 87, 88, 89, 90, 93, 94, 95, 96, 99

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

The sequence is the interleaving of A047233 with A047270. - Guenther Schrack, Feb 15 2019

Links

• Guenther Schrack, Table of n, a(n) for n = 1..10010
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Crossrefs

Cf. A047233, A047246, A047270.

Complement: A047239.

Programs

Magma

[n : n in [0..150] | n mod 6 in [0, 3, 4, 5]]; // Wesley Ivan Hurt, Jun 02 2016

Maple

A047250:=n->(6*n-3+I^(2*n)-(1+I)*I^(-n)-(1-I)*I^n)/4: seq(A047250(n), n=1..100); # Wesley Ivan Hurt, Jun 02 2016

Mathematica

Select[Range[0,100], MemberQ[{0,3,4,5}, Mod[#,6]]&] (* or *) LinearRecurrence[{1,0,0,1,-1}, {0,3,4,5,6}, 60] (* Harvey P. Dale, Apr 01 2013 *)

PARI

my(x='x+O('x^70)); concat([0], Vec(x^2*(3+x+x^2+x^3)/((1+x)*(1+x^2)*(1-x)^2))) \\ G. C. Greubel, Feb 16 2019

Sage

a=(x^2*(3+x+x^2+x^3)/((1+x)*(1+x^2)*(1-x)^2)).series(x, 72).coefficients(x, sparse=False); a[1:] # G. C. Greubel, Feb 16 2019

Formula

G.f.: x^2*(3+x+x^2+x^3)/((1+x)*(1+x^2)*(1-x)^2). - R. J. Mathar, Oct 08 2011

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

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

a(n) = (6*n - 3 + i^(2*n) - (1+i)*i^(-n) - (1-i)*i^n)/4 where i=sqrt(-1).

a(2*k) = A047270(k), a(2*k-1) = A047233(k). (End)

E.g.f.: (2 - sin(x) - cos(x) + (3*x - 2)*sinh(x) + (3*x - 1)*cosh(x))/2. - Ilya Gutkovskiy, Jun 02 2016

From Guenther Schrack, Feb 15 2019: (Start)

a(n) = (6*n - 3 + (-1)^n - 2*(-1)^(n*(n-1)/2))/4.

a(n) = a(n-4) + 6, a(1)=0, a(2)=3, a(3)=4, a(4)=5, for n > 4.

a(-n) = -A047246(n+2). (End)

Sum_{n>=2} (-1)^n/a(n) = 2*log(2)/3 - Pi/(6*sqrt(3)). - Amiram Eldar, Dec 17 2021