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A265667 Permutation of nonnegative integers: a(n) = n + floor(n/3)*(-1)^(n mod 3). 11
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 (list; graph; refs; listen; history; text; internal format)
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
Peter Lynch and Michael Mackey, Parity and Partition of the Rational Numbers, arXiv:2205.00565 [math.NT], 2022. See set F p. 4.
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
(Sage) [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. 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
KEYWORD
nonn,easy
AUTHOR
Bruno Berselli, Dec 12 2015 - based on an idea by Paul Curtz.
STATUS
approved

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Last modified June 18 09:40 EDT 2024. Contains 373472 sequences. (Running on oeis4.)