A236343 Expansion of (1 - x + 2*x^2 - x^3) / ((1 - x)^2 * (1 - x^3)) in powers of x.

1, 1, 3, 5, 6, 9, 12, 14, 18, 22, 25, 30, 35, 39, 45, 51, 56, 63, 70, 76, 84, 92, 99, 108, 117, 125, 135, 145, 154, 165, 176, 186, 198, 210, 221, 234, 247, 259, 273, 287, 300, 315, 330, 344, 360, 376, 391, 408, 425, 441, 459, 477, 494, 513, 532, 550, 570, 590

Offset

0,3

Comments

The sequence is a quasi-polynomial sequence.

Given a sequence of Laurent polynomials defined by b(n) = (b(n-2)^2 - b(n-1)*b(n-3) * 2/x) / b(n-4), b(-2) = x, b(-4) = -b(-3) = -b(-1) = 1. Then the denominator of b(n) is x^a(n).

Links

G. C. Greubel, Table of n, a(n) for n = 0..2500

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

Formula

0 = a(n)*(a(n+2) + a(n+3)) + a(n+1)*(-2*a(n+2) - a(n+3) + a(n+4)) + a(n+2)*(a(n+2) - 2*a(n+3) + a(n+4)) for all n in Z.

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

Second difference is period 3 sequence [2, 0, -1, ...].

a(n) = 2*a(n-3) + a(n-6) + 3 = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5).

a(-6-n) = A236337(n).

From Peter Bala, Feb 11 2019: (Start)

a(3*n) = (1/2)*(n + 1)*(3*n + 2);

a(3*n+1) = (1/2)*(n + 1)*(3*n + 4) - 1;

a(3*n+2) = (1/2)*(n + 1)*(3*n + 6). (End)

Example

G.f. = 1 + x + 3*x^2 + 5*x^3 + 6*x^4 + 9*x^5 + 12*x^6 + 14*x^7 + 18*x^8 + ...

Maple

seq(coeff(series((1-x+2*x^2-x^3)/((1-x)^2*(1-x^3)), x, n+1), x, n), n = 0 .. 60); # Muniru A Asiru, Feb 12 2019

Mathematica

CoefficientList[Series[(1-x+2*x^2-x^3)/((1-x)^2*(1-x^3)), {x, 0, 60}], x] (* G. C. Greubel, Aug 07 2018 *)

Prog

(PARI) {a(n) = (n * (n+5) + [6, 0, 4][n%3 + 1]) / 6};
(PARI) {a(n) = if( n<0, polcoeff( x^2 * (-1 + 2*x - x^2 + x^3) / ((1 - x)^2 * (1 - x^3)) + x * O(x^-n), -n), polcoeff( (1 - x + 2*x^2 - x^3) / ((1 - x)^2 * (1 - x^3)) + x * O(x^n), n))};
(Magma) m:=60; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-x+2*x^2-x^3)/((1-x)^2*(1-x^3)))); // G. C. Greubel, Aug 07 2018
(Sage) ((1-x+2*x^2-x^3)/((1-x)^2*(1-x^3))).series(x, 60).coefficients(x, sparse=False) # G. C. Greubel, Feb 12 2019

Crossrefs

Cf. A236337. Trisections are A000326, A095794, A045943.

Sequence in context: A323115 A285377 A192577 * A325420 A168063 A039873

Adjacent sequences: A236340 A236341 A236342 * A236344 A236345 A236346

Keyword

nonn,easy

Author

Michael Somos, Jan 22 2014

Status

approved