A154571 Numbers that are congruent to {0, 3, 4, 5, 7, 8} mod 12.

0, 3, 4, 5, 7, 8, 12, 15, 16, 17, 19, 20, 24, 27, 28, 29, 31, 32, 36, 39, 40, 41, 43, 44, 48, 51, 52, 53, 55, 56, 60, 63, 64, 65, 67, 68, 72, 75, 76, 77, 79, 80, 84, 87, 88, 89, 91, 92, 96, 99, 100, 101, 103, 104, 108, 111, 112, 113, 115, 116, 120, 123, 124

Offset: 1

Author

Omar E. Pol, Jan 12 2009

Keywords

• easy
• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,1,-1).

Crossrefs

Cf. A113829.

Programs

Magma

[n : n in [0..150] | n mod 12 in [0, 3, 4, 5, 7, 8]]; // Wesley Ivan Hurt, May 29 2016

Maple

A154571:=n->(12*n-15-cos(n*Pi)-5*cos(n*Pi/3)-sqrt(3)*(2*cos((1-4*n)*Pi/6)-3*sin(n*Pi/3)))/6: seq(A154571(n), n=1..100); # Wesley Ivan Hurt, Jun 16 2016

Mathematica

LinearRecurrence[{1, 0, 0, 0, 0, 1, -1}, {0, 3, 4, 5, 7, 8, 12}, 50] (* G. C. Greubel, May 29 2016 *)

Formula

From Chai Wah Wu, May 29 2016: (Start)

a(n) = a(n-1) + a(n-6) - a(n-7) for n>7.

G.f.: x^2*(4*x^5 + x^4 + 2*x^3 + x^2 + x + 3)/(x^7 - x^6 - x + 1). (End)

From Wesley Ivan Hurt, Jun 16 2016: (Start)

a(n) = (12*n - 15 - cos(n*Pi) - 5*cos(n*Pi/3) - sqrt(3)*(2*cos((1-4*n)*Pi/6) - 3*sin(n*Pi/3)))/6.

a(6k) = 12k-4, a(6k-1) = 12k-5, a(6k-2) = 12k-7, a(6k-3) = 12k-8, a(6k-4) = 12k-9, a(6k-5) = 12k-12. (End)

Sum_{n>=2} (-1)^n/a(n) = (15-8*sqrt(3))*Pi/72 + log(2)/4. - Amiram Eldar, Dec 31 2021