A265667 Permutation of nonnegative integers: a(n) = n + floor(n/3)*(-1)^(n mod 3).

0, 1, 2, 4, 3, 6, 8, 5, 10, 12, 7, 14, 16, 9, 18, 20, 11, 22, 24, 13, 26, 28, 15, 30, 32, 17, 34, 36, 19, 38, 40, 21, 42, 44, 23, 46, 48, 25, 50, 52, 27, 54, 56, 29, 58, 60, 31, 62, 64, 33, 66, 68, 35, 70, 72, 37, 74, 76, 39, 78, 80, 41, 82, 84, 43, 86, 88, 45

Offset

0,3

Comments

The inverse permutation is given by P(n) = A006368(n-1) + 1, for n >= 1, and P(0) = 0. - Wolfdieter Lang, Sep 21 2021

This permutation is given by A006369(n-1) + 1, with A006369(-1) = -1. Observed by Kevin Ryde. - Wolfdieter Lang, Sep 22 2021

Links

Bruno Berselli, Table of n, a(n) for n = 0..1000

Peter Lynch and Michael Mackey, Parity and Partition of the Rational Numbers, arXiv:2205.00565 [math.NT], 2022. See set F p. 4.

Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,-1).

Index entries for permutations of the positive (or nonnegative) integers.

Formula

G.f.: x*(1 + 2*x + 4*x^2 + x^3 + 2*x^4) / ((1 - x)^2*(1 + x + x^2)^2).

a(n) = 2*a(n-3) - a(n-6).

a(3*k) = 4*k;

a(3*k+1) = 2*k+1, hence a(3*k+1) = a(3*k)/2 + 1;

a(3*k+2) = 4*k+2, hence a(3*k+2) = 2*a(3*k+1) = a(3*k) + 2.

Sum_{i=0..n} a(i) = A008738(A032793(n+1)).

Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/8 + log(2)/4. - Amiram Eldar, Mar 30 2023

Example

-------------------------------------------------------------------------
0, 1, 2, 3,  4, 5, 6,  7,  8,  9, 10, 11, 12, 13, 14, 15, 16, 17, 18, ...
+  +  +  +   +  +  +   +   +   +   +   +   +   +   +   +   +   +   +
0, 0, 0, 1, -1, 1, 2, -2,  2,  3, -3,  3,  4, -4,  4,  5, -5,  5,  6, ...
-------------------------------------------------------------------------
0, 1, 2, 4,  3, 6, 8,  5, 10, 12,  7, 14, 16,  9, 18, 20, 11, 22, 24, ...
-------------------------------------------------------------------------

Mathematica

Table[n + Floor[n/3] (-1)^Mod[n, 3], {n, 0, 70}]

Prog

(SageMath) [n+floor(n/3)*(-1)^mod(n, 3) for n in (0..70)]
(Magma) [n+Floor(n/3)*(-1)^(n mod 3): n in [0..70]];

Crossrefs

Cf. A001477, A006368, A006369, A008738, A032793.

Cf. A064455: n+floor(n/2)*(-1)^(n mod 2).

Cf. A265888: n+floor(n/4)*(-1)^(n mod 4).

Cf. A265734: n+floor(n/5)*(-1)^(n mod 5).

Sequence in context: A272904 A233342 A120233 * A338743 A358267 A280866

Adjacent sequences: A265664 A265665 A265666 * A265668 A265669 A265670

Keyword

nonn,easy

Author

Bruno Berselli, Dec 12 2015 - based on an idea by Paul Curtz

Status

approved