A047251 Numbers that are congruent to {1, 3, 4, 5} (mod 6).

1, 3, 4, 5, 7, 9, 10, 11, 13, 15, 16, 17, 19, 21, 22, 23, 25, 27, 28, 29, 31, 33, 34, 35, 37, 39, 40, 41, 43, 45, 46, 47, 49, 51, 52, 53, 55, 57, 58, 59, 61, 63, 64, 65, 67, 69, 70, 71, 73, 75, 76, 77, 79, 81, 82, 83, 85, 87, 88, 89, 91, 93, 94, 95, 97, 99

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

"Polyrhythmic Sequence" P(2,3): numbers congruent to 1 (mod 2) and 1 (mod 3). (See A267027 for definition and description.) - Bob Selcoe, Jan 12 2016

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).

Crossrefs

Cf. A016777, A047270, A267027.

Programs

Magma

[n: n in [0..150]|n mod 6 in {1,3,4,5}]; // Vincenzo Librandi, Jan 12 2016

Maple

A047251:=n->(-2-(-I)^n-I^n+6*n)/4: seq(A047251(n), n=1..100); # Wesley Ivan Hurt, May 31 2016

Mathematica

Select[Range[0, 200], MemberQ[{1, 3, 4, 5}, Mod[#, 6]] &] (* Vincenzo Librandi, Jan 12 2016 *)
LinearRecurrence[{2,-2,2,-1},{1,3,4,5},70] (* Harvey P. Dale, Feb 27 2024 *)

PARI

a(n) = (-2-(-I)^n-I^n+6*n)/4 \\ Colin Barker, Oct 19 2015

PARI

Vec(x*(x^3+x+1)/((x-1)^2*(x^2+1)) + O(x^100)) \\ Colin Barker, Oct 19 2015

Formula

From R. J. Mathar, Oct 08 2011: (Start)

a(n) = 3*n/2 - 1/2 - cos(Pi*n/2)/2.

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

a(n) = (-2 - (-i)^n - i^n + 6n)/4, with i=sqrt(-1). - Colin Barker, Oct 19 2015

From Wesley Ivan Hurt, May 31 2016: (Start)

a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) for n>4.

a(2k) = A047270(k), a(2k-1) = A016777(k-1) for n>0. (End)

Sum_{n>=1} (-1)^(n+1)/a(n) = 5*sqrt(3)*Pi/36 - log(2)/3 + log(3)/4. - Amiram Eldar, Dec 17 2021