A350668 Numbers congruent to 2, 4, and 6 modulo 9: positions of 2 in A159955.

2, 4, 6, 11, 13, 15, 20, 22, 24, 29, 31, 33, 38, 40, 42, 47, 49, 51, 56, 58, 60, 65, 67, 69, 74, 76, 78, 83, 85, 87, 92, 94, 96, 101, 103, 105, 110, 112, 114, 119, 121, 123, 128, 130, 132, 137, 139, 141, 146, 148

Offset

0,1

Comments

This sequence, together with A350666 and A350667, gives a 3-set partition of the nonnegative integers.

This sequence {a(n)}_{n>=0} gives the indices of the row sequences of array A = A347834, that are modulo 6 periodic with period length 3, namely

{A347834(a(n), m) mod 6}_{m >= 0} = {repeat(3, 1, 5)}.

Links

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Formula

A159955(a(n)) = 2.

Trisection: a(3*k) = 2 + 9*k, a(3*k + 1) = 4 + 9*k, and a(3*k + 3) = 6 + 9*k, for k >= 0.

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

a(n) = 1 + 3*n + cos(2*n*Pi/3) + sin(2*n*Pi/3)/sqrt(3). - Stefano Spezia, Jan 30 2022

a(n) = 1 + 3*n + S(2*n, 1) = 1+3*n+A057078(n), with the Chebyshev S polynomials from A049310, using the partial fraction decomposition of the g.f., or the previous formula.

Example

Rows of array {A347834(a(n), m)}_{m >= 0}, with modulo 6 congruence:
n = 0: row 2: {3, 13, 53, 213, 853, 3413, 13653, ...} mod 6 = {repeat(3, 1, 5)},
n = 1: row 4: {9, 37, 149, 597, 2389, 9557, ...} (mod 6) = {repeat(3, 1, 5)},
...

Mathematica

Select[Range[0, 150], MemberQ[{2, 4, 6}, Mod[#, 9]] &] (* Amiram Eldar, Jan 29 2022 *)
LinearRecurrence[{1, 0, 1, -1}, {2, 4, 6, 11}, 80] (* Harvey P. Dale, Jul 12 2024 *)

Crossrefs

Cf. A049310, A159955, A347834, A350666, A350667.

Sequence in context: A015903 A224860 A327309 * A078198 A171865 A372622

Adjacent sequences: A350665 A350666 A350667 * A350669 A350670 A350671

Keyword

nonn,easy

Author

Wolfdieter Lang, Jan 29 2022

Status

approved