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%I #30 Dec 12 2023 07:46:01
%S 1,0,2,4,3,5,7,6,8,10,9,11,13,12,14,16,15,17,19,18,20,22,21,23,25,24,
%T 26,28,27,29,31,30,32,34,33,35,37,36,38,40,39,41,43,42,44,46,45,47,49,
%U 48,50,52,51,53,55,54,56,58,57,59,61,60,62,64,63,65,67
%N Expansion of (1 - x + 2*x^2 + x^3) / ((1 - x) * (1 - x^3)) in powers of x.
%C An order 2 permutation of nonnegative integers.
%H Vincenzo Librandi, <a href="/A236348/b236348.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1,-1).
%F G.f.: (1 - x + 2*x^2 + x^3) / ((1 - x) * (1 - x^3)).
%F First difference is period 3 sequence [-1, 2, 2, ...].
%F a(n) = a(n-1) + a(n-3) - a(n-4). a(4-n) = 4-a(n).
%F 0 = a(n)*(-a(n+1) + a(n+3)) + a(n+1)*(a(n+1) - a(n+2)) + a(n+2)*(a(n+2) - a(n+3)) for all n in Z.
%F a(n) = A143097(n) if n>1.
%F a(n) = n - 1 + mod(n-1, 3). - _Wesley Ivan Hurt_, Aug 21 2014
%F a(n) = n + (2/sqrt(3))*sin(2*(n+1)*Pi/3). - _Wesley Ivan Hurt_, Sep 26 2017
%F Sum_{n>=2} (-1)^n/a(n) = 2*Pi/(3*sqrt(3)) + log(2)/3 - 1. - _Amiram Eldar_, Sep 10 2023
%e G.f. = 1 + 2*x^2 + 4*x^3 + 3*x^4 + 5*x^5 + 7*x^6 + 6*x^7 + 8*x^8 + 10*x^9 + ...
%t Table[n - 1 + Mod[n - 1, 3], {n, 0, 100}] (* _Wesley Ivan Hurt_, Aug 21 2014 *)
%t LinearRecurrence[{1, 0, 1, -1}, {1, 0, 2, 4}, 80] (* or *) CoefficientList[Series[(1 - x + 2 x^2 + x^3) / ((1 - x) (1 -x^3)), {x, 0, 80}], x] (* _Vincenzo Librandi_, Sep 28 2017 *)
%o (PARI) {a(n) = (n-1) % 3 + n-1 }
%o (Magma) [n-1+((n-1) mod 3) : n in [0..100]]; // _Wesley Ivan Hurt_, Aug 21 2014
%o (Magma) I:=[1,0,2,4]; [n le 4 select I[n] else Self(n-1)+Self(n-3)-Self(n-4): n in [1..80]]; // _Vincenzo Librandi_, Sep 28 2017
%Y Cf. A143097.
%K nonn,easy
%O 0,3
%A _Michael Somos_, Jan 23 2014
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