A047269 Numbers that are congruent to {0, 1, 2, 5} mod 6.

0, 1, 2, 5, 6, 7, 8, 11, 12, 13, 14, 17, 18, 19, 20, 23, 24, 25, 26, 29, 30, 31, 32, 35, 36, 37, 38, 41, 42, 43, 44, 47, 48, 49, 50, 53, 54, 55, 56, 59, 60, 61, 62, 65, 66, 67, 68, 71, 72, 73, 74, 77, 78, 79, 80, 83, 84, 85, 86, 89, 90, 91, 92, 95, 96, 97

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, 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. A007310, A047238.

Programs

Magma

I:=[0, 1, 2, 5, 6]; [n le 5 select I[n] else Self(n-1)+Self(n-4)-Self(n-5): n in [1..70]]; // Vincenzo Librandi, May 15 2012

Maple

A047269:=n->(-7+(-1)^n+(1+I)*(-I)^n+(1-I)*I^n+6*n)/4: seq(A047269(n), n=1..100); # Wesley Ivan Hurt, May 22 2016

Mathematica

Select[Range[0,4000], MemberQ[{0,1,2,5}, Mod[#,6]]&] (* Vincenzo Librandi, May 15 2012 *)
LinearRecurrence[{1,0,0,1,-1},{0,1,2,5,6},80] (* Harvey P. Dale, Jun 21 2022 *)

PARI

x='x+O('x^100); concat(0, Vec(x^2*(1+x+3*x^2+x^3)/((1-x)^2*(1+x)*(1+x^2)))) \\ Altug Alkan, Dec 24 2015

Formula

From Colin Barker, May 14 2012: (Start)

a(n) = (-7+(-1)^n+(1+i)*(-i)^n+(1-i)*i^n+6*n)/4 where i=sqrt(-1).

G.f.: x^2*(1+x+3*x^2+x^3)/((1-x)^2*(1+x)*(1+x^2)). (End)

a(n) = a(n-1) + a(n-4) - a(n-5) for n>5. - Vincenzo Librandi, May 15 2012

a(2n) = A007310(n), a(2n-1) = A047238(n). - Wesley Ivan Hurt, May 22 2016

Sum_{n>=2} (-1)^n/a(n) = log(3)/4 + 2*log(2)/3 - sqrt(3)*Pi/36. - Amiram Eldar, Dec 16 2021

a(n) = n - 1 + 2*floor(n/4). - Taras Goy, Jan 03 2025