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

1,2

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).

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

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 *)

Prog

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

Crossrefs

Cf. A047233, A047246, A047270.

Complement: A047239.

Sequence in context: A364064 A338321 A220844 * A081944 A129948 A050414

Adjacent sequences: A047247 A047248 A047249 * A047251 A047252 A047253

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved