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

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

0,2

Comments

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)}.

Links

Table of n, a(n) for n=0..49.

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

Formula

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).

Example

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)},
...

Mathematica

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 *)

Crossrefs

Cf. A159955, A347834, A350667, A350668.

Sequence in context: A155732 A218851 A211184 * A327307 A029606 A247283

Adjacent sequences: A350663 A350664 A350665 * A350667 A350668 A350669

Keyword

nonn,easy

Author

Wolfdieter Lang, Jan 29 2022

Status

approved