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

%I A265667 #44 Jul 09 2025 04:40:33
%S A265667 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,
%T A265667 34,36,19,38,40,21,42,44,23,46,48,25,50,52,27,54,56,29,58,60,31,62,64,
%U A265667 33,66,68,35,70,72,37,74,76,39,78,80,41,82,84,43,86,88,45
%N A265667 Permutation of nonnegative integers: a(n) = n + floor(n/3)*(-1)^(n mod 3).
%C A265667 The inverse permutation is given by P(n) = A006368(n-1) + 1, for n >= 1, and P(0) = 0. - _Wolfdieter Lang_, Sep 21 2021
%C A265667 This permutation is given by A006369(n-1) + 1, with A006369(-1) = -1. Observed by _Kevin Ryde_. - _Wolfdieter Lang_, Sep 22 2021
%H A265667 Bruno Berselli, <a href="/A265667/b265667.txt">Table of n, a(n) for n = 0..1000</a>
%H A265667 Peter Lynch and Michael Mackey, <a href="https://arxiv.org/abs/2205.00565">Parity and Partition of the Rational Numbers</a>, arXiv:2205.00565 [math.NT], 2022. See set F p. 4.
%H A265667 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,2,0,0,-1).
%H A265667 <a href="/index/Per#IntegerPermutation">Index entries for permutations of the positive (or nonnegative) integers</a>.
%F A265667 G.f.: x*(1 + 2*x + 4*x^2 + x^3 + 2*x^4) / ((1 - x)^2*(1 + x + x^2)^2).
%F A265667 a(n) = 2*a(n-3) - a(n-6).
%F A265667 a(3*k)   = 4*k;
%F A265667 a(3*k+1) = 2*k+1, hence a(3*k+1) = a(3*k)/2 + 1;
%F A265667 a(3*k+2) = 4*k+2, hence a(3*k+2) = 2*a(3*k+1) = a(3*k) + 2.
%F A265667 Sum_{i=0..n} a(i) = A008738(A032793(n+1)).
%F A265667 Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/8 + log(2)/4. - _Amiram Eldar_, Mar 30 2023
%e A265667 -------------------------------------------------------------------------
%e A265667 0, 1, 2, 3,  4, 5, 6,  7,  8,  9, 10, 11, 12, 13, 14, 15, 16, 17, 18, ...
%e A265667 +  +  +  +   +  +  +   +   +   +   +   +   +   +   +   +   +   +   +
%e A265667 0, 0, 0, 1, -1, 1, 2, -2,  2,  3, -3,  3,  4, -4,  4,  5, -5,  5,  6, ...
%e A265667 -------------------------------------------------------------------------
%e A265667 0, 1, 2, 4,  3, 6, 8,  5, 10, 12,  7, 14, 16,  9, 18, 20, 11, 22, 24, ...
%e A265667 -------------------------------------------------------------------------
%t A265667 Table[n + Floor[n/3] (-1)^Mod[n, 3], {n, 0, 70}]
%o A265667 (Sage) [n+floor(n/3)*(-1)^mod(n,3) for n in (0..70)]
%o A265667 (Magma) [n+Floor(n/3)*(-1)^(n mod 3): n in [0..70]];
%Y A265667 Cf. A001477, A006368, A006369, A008738, A032793.
%Y A265667 Cf. A064455: n+floor(n/2)*(-1)^(n mod 2).
%Y A265667 Cf. A265888: n+floor(n/4)*(-1)^(n mod 4).
%Y A265667 Cf. A265734: n+floor(n/5)*(-1)^(n mod 5).
%K A265667 nonn,easy
%O A265667 0,3
%A A265667 _Bruno Berselli_, Dec 12 2015 - based on an idea by _Paul Curtz_