A281899 a(n) = n + 6*floor(n/3).

0, 1, 2, 9, 10, 11, 18, 19, 20, 27, 28, 29, 36, 37, 38, 45, 46, 47, 54, 55, 56, 63, 64, 65, 72, 73, 74, 81, 82, 83, 90, 91, 92, 99, 100, 101, 108, 109, 110, 117, 118, 119, 126, 127, 128, 135, 136, 137, 144, 145, 146, 153, 154, 155, 162, 163, 164, 171, 172, 173, 180, 181, 182, 189

Offset: 0

Author

Bruno Berselli, Feb 06 2017

Keywords

• nonn
• easy

Comments

Equivalently, numbers that are congruent to {0, 1, 2} mod 9.

Also numbers m such that floor(m/3) = 3*floor(m/9).

The n-th term is 3*n, 3*n-2 or 3*n-4.

For n > 0, numbers k such that 3 | floor(k/3). - Wesley Ivan Hurt, Dec 01 2020

Links

• Bruno Berselli, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Crossrefs

Cf. A002264.

Subsequence of A060464 and A248375.

The first differences are in A105395.

Cf. similar sequences with formula n+i*floor(n/3): A004773 (i=1), A047217 (i=2), A047240 (i=3), A047354 (i=4), A047469 (i=5), this sequence (i=6).

Cf. numbers that are congruent to {0, 1, 2} mod j: the sequences are listed in the previous row for j = 4..9, respectively.

Programs

Magma

[n+6*(n div 3): n in [0..70]];

Maple

A281899:=n->n+6*floor(n/3): seq(A281899(n), n=0..100); # Wesley Ivan Hurt, Feb 09 2017

Mathematica

Table[n + 6 Floor[n/3], {n, 0, 70}]
LinearRecurrence[{1,0,1,-1},{0,1,2,9},90] (* Harvey P. Dale, Feb 25 2018 *)

Maxima

makelist(n+6*floor(n/3), n, 0, 70);

PARI

a(n)=n\3*6 + n \\ Charles R Greathouse IV, Feb 07 2017

Python

[n+6*int(n/3) for n in range(70)]

Sage

[n+6*floor(n/3) for n in range(70)]

Formula

G.f.: x*(1 + x + 7*x^2)/((1 - x)^2*(1 + x + x^2)).

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

a(n) = 3*n - 2*(n mod 3). In general, n + 3*h*floor(n/3) = (h+1)*n - h*(n mod 3).

a(n) + a(n+s) = a(2*n+s-1) + 1, where s is nonnegative and not divisible by 3. Example: for s=14, a(n) + a(n+14) = a(2*n+13) + 1; for n=3, a(3) + a(17) = a(19) + 1 = 9 + 47 = 55 + 1 = 56.

a(6*k+r) = 18*k + a(r), where 0 <= r <= 5.

a(n) = 7*A002264(n) + A002264(n+1) + A002264(n+2).