A029739 Numbers that are congruent to {1, 3, 4} mod 6.

1, 3, 4, 7, 9, 10, 13, 15, 16, 19, 21, 22, 25, 27, 28, 31, 33, 34, 37, 39, 40, 43, 45, 46, 49, 51, 52, 55, 57, 58, 61, 63, 64, 67, 69, 70, 73, 75, 76, 79, 81, 82, 85, 87, 88, 91, 93, 94, 97, 99, 100, 103, 105, 106, 109, 111, 112, 115, 117, 118, 121, 123, 124

Offset: 1

Author

Patrick De Geest

Keywords

• nonn
• easy

Links

• Patrick De Geest, More palindromic products of integer sequences: Three consecutive palindromes.
• Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Programs

Magma

[n : n in [0..150] | n mod 6 in {1, 3, 4}]; // Vincenzo Librandi, Dec 29 2010

Maple

A029739:=n->2*(3*n-2-cos(2*n*Pi/3))/3: seq(A029739(n), n=1..100); # Wesley Ivan Hurt, Jun 11 2016

Mathematica

Select[Range[0, 202], MemberQ[{1, 3, 4}, Mod[#, 6]] &] (* and *) Join[{1}, Accumulate[Total /@ CellularAutomaton[65, {1, 1, 0, 0, 1, 0}, 100]]] (* Vladimir Joseph Stephan Orlovsky, Feb 11 2012 *)
LinearRecurrence[{1,0,1,-1},{1,3,4,7},80] (* Harvey P. Dale, Aug 21 2021 *)

Formula

G.f.: x*(2*x+1)*(x^2+1)/((1+x+x^2)*(x-1)^2). - R. J. Mathar, Aug 24 2011

From Wesley Ivan Hurt, Jun 11 2016: (Start)

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

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

a(3k) = 6k-2, a(3k-1) = 6k-3, a(3k-2) = 6k-5. (End)

Sum_{n>=1} (-1)^(n+1)/a(n) = (3+2*sqrt(3))*Pi/36 + log(2+sqrt(3))/(2*sqrt(3)) - log(2)/6. - Amiram Eldar, Dec 16 2021