A350667 -id:A350667 - cp's OEIS frontend

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

Wolfdieter Lang, Jan 29 2022

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)}.

			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)},
...

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).

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

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 *)

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.

A350666 Numbers congruent to 0, 5, and 7 modulo 9: positions of 0 in A159955.

Original entry on oeis.org

0, 5, 7, 9, 14, 16, 18, 23, 25, 27, 32, 34, 36, 41, 43, 45, 50, 52, 54, 59, 61, 63, 68, 70, 72, 77, 79, 81, 86, 88, 90, 95, 97, 99, 104, 106, 108, 113, 115, 117, 122, 124, 126, 131, 133, 135, 140, 142, 144, 149

Offset: 0

Wolfdieter Lang, Jan 29 2022

This sequence, together with A350667 and A350668, gives a 3-set partition of the nonnegative integers.

This sequence {a(n)}, for n >= 1, 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(0, 3, 1)}.

			Rows of array {A347834(a(n), m)}_{m>=0}, with modulo 6 congruence:
n = 1: row 5: {11, 45, 181, 725, 2901, 11605,...} mod 6 = {5, 3, 1, 5, 3, 1, ...},
n = 2: row 7: {17, 69, 277, 1109, 4437, 17749, ...} mod 6 = {repeat(5, 3, 1)},
...

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

Cf. A159955, A347834, A350667, A350668.

Select[Range[0, 150], MemberQ[{0, 5, 7}, Mod[#, 9]] &] (* Amiram Eldar, Jan 29 2022 *)
Table[1 + 3n - ChebyshevU[n,-1/2],{n,0,49}] (* Stefano Spezia, Jan 30 2022 *)

A159955(a(n)) = 0.
Trisection: a(3*k) = 9*k, a(3*k+1) = 5 + 9*k, and a(3*k+2) = 7 + 9*k, for k >= 0.
G.f.: x*(5 + 2*x + 2*x^2)/((1 - x)*(1 - x^3)).
a(n) = 1 + 3*n - U(n, -1/2) = 1+3*n-A049347(n), where U(n, x) is a Chebyshev U-polynomial. - Stefano Spezia, Jan 30 2022
a(n) = 1 + 3*n - (2/sqrt(3))*sin(2*(n+1)*Pi/3) (from the previous formula).

cp's OEIS Frontend

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

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