A236348 Expansion of (1 - x + 2*x^2 + x^3) / ((1 - x) * (1 - x^3)) in powers of x.

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, 26, 28, 27, 29, 31, 30, 32, 34, 33, 35, 37, 36, 38, 40, 39, 41, 43, 42, 44, 46, 45, 47, 49, 48, 50, 52, 51, 53, 55, 54, 56, 58, 57, 59, 61, 60, 62, 64, 63, 65, 67

Offset: 0

Author

Michael Somos, Jan 23 2014

Keywords

• nonn
• easy

Comments

An order 2 permutation of nonnegative integers.

Examples

			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 + ...
		

Links

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

Crossrefs

Cf. A143097.

Programs

Magma

[n-1+((n-1) mod 3) : n in [0..100]]; // Wesley Ivan Hurt, Aug 21 2014

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

Mathematica

Table[n - 1 + Mod[n - 1, 3], {n, 0, 100}] (* Wesley Ivan Hurt, Aug 21 2014 *)
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 *)

PARI

{a(n) = (n-1) % 3 + n-1 }

Formula

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

First difference is period 3 sequence [-1, 2, 2, ...].

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

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.

a(n) = A143097(n) if n>1.

a(n) = n - 1 + mod(n-1, 3). - Wesley Ivan Hurt, Aug 21 2014

a(n) = n + (2/sqrt(3))*sin(2*(n+1)*Pi/3). - Wesley Ivan Hurt, Sep 26 2017

Sum_{n>=2} (-1)^n/a(n) = 2*Pi/(3*sqrt(3)) + log(2)/3 - 1. - Amiram Eldar, Sep 10 2023