(MAGMAMagma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((-1-x+x^2)/(1+2*x+2*x^2-x^3))); // G. C. Greubel, Aug 07 2018

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G. C. Greubel, <a href="/indexA233831/Recb233831.txt">Index entries Table of n, a(n) for linear recurrences with constant coefficientsn = 0..1000</a>, signature (-2, -2, 1).

<a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (-2, -2, 1).

CoefficientList[Series[(-1-x+x^2)/(1+2*x+2*x^2-x^3), {x, 0, 50}], x] (* G. C. Greubel, Aug 07 2018 *)

(MAGMA) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((-1-x+x^2)/(1+2*x+2*x^2-x^3))); // G. C. Greubel, Aug 07 2018

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<a href="/index/Rec">Index to sequences with entries for linear recurrences with constant coefficients</a>, signature (-2, -2, 1).

<a href="/index/Rea#recLCCRec">Index to sequences with linear recurrences with constant coefficients</a>, signature (-2, -2, 1).

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allocated for Michael Somos

a(n) = -2*a(n-1) -2*a(n-2) + a(n-3). a(0) = -1, a(1) = 1, a(2) = 1.

-1, 1, 1, -5, 9, -7, -9, 41, -71, 51, 81, -335, 559, -367, -719, 2731, -4391, 2601, 6311, -22215, 34409, -18077, -54879, 180321, -268961, 122401, 473441, -1460645, 2096809, -798887, -4056489, 11807561, -16301031, 4930451, 34548721, -95259375, 126351759

0,4

<a href="/index/Rea#recLCC">Index to sequences with linear recurrences with constant coefficients</a>, signature (-2, -2, 1).

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

a(-n) = A233828(n).

a(n) - a(n-1) = -2 * (-1)^n * A078004(n).

a(n)^2 - a(n-1) * a(n+1) = -2 * (-1)^n * A078054(n-1).

G.f. = -1 + x + x^2 - 5*x^3 + 9*x^4 - 7*x^5 - 9*x^6 + 41*x^7 - 71*x^8 + ...

(PARI) {a(n) = if( n<0, polcoeff( (-1 +3*x + x^2) / (1 - 2*x - 2*x^2 - x^3) + x * O(x^-n), -n), polcoeff( (-1 - x + x^2) / (1 + 2*x + 2*x^2 - x^3) + x * O(x^n), n))}

Cf. A078004, A078054, A233828.

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Michael Somos, Dec 16 2013

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