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A236343 Expansion of (1 - x + 2*x^2 - x^3) / ((1 - x)^2 * (1 - x^3)) in powers of x. 2
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified October 23 01:49 EDT 2020. Contains 337962 sequences. (Running on oeis4.)