A350667 Numbers congruent to 1, 3, and 8 modulo 9: positions of 1 in A159955.

1, 3, 8, 10, 12, 17, 19, 21, 26, 28, 30, 35, 37, 39, 44, 46, 48, 53, 55, 57, 62, 64, 66, 71, 73, 75, 80, 82, 84, 89, 91, 93, 98, 100, 102, 107, 109, 111, 116, 118, 120, 125, 127, 129, 134, 136, 138, 143, 145, 147

Offset: 0

Author

Wolfdieter Lang, Jan 29 2022

Keywords

• nonn
• easy

Comments

This sequence, together with A350666 and A350668, gives a 3-set partition of the nonnegative integers.

This sequence {a(n)}A347834,%20that%20are%20modulo%206%20periodic%20with%20period%20length%203,%20namely%20%7BA347834(a(n),%20m)%20mod%206%7D">{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(1, 5, 3)}.

Examples

			Rows of array {A347834(a(n), m)}_{m>=0}, with modulo 6 congruence:
n = 0: row 1: {1, 5, 21, 85, 341, 1365, 5461, ...} mod 6 = {repeat(1, 5, 3)},
n = 1: row 3: {7, 29, 117, 469, 1877, 7509, ...} mod 6 = {repeat(1, 5, 3)},
...
		

Links

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

Crossrefs

Cf. A049310, A055264, A159955, A347834, A350666, A350668.

Programs

Mathematica

Select[Range[0, 150], MemberQ[{1, 3, 8}, Mod[#, 9]] &] (* Amiram Eldar, Jan 29 2022 *)

Formula

A159955(a(n)) = 1.

Trisection: a(3*k) = 1 + 9*k, a(3*k+1) = 3 + 9*k, and a(3*k+3) = 8 + 9*k, for k >= 0.

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

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

a(n) = 1 + 3*n - S(n-1,-1), with S(-1, x) = 0, with the Chebyshev S polynomials from A049310. From the g.f., or from the previous formula (see also Spezia's formula in A350666).