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Expansion of x * (96 - 816*x) / ((1 - x) * (1 - 1442*x + x^2)) in powers of x.
1

%I #17 Sep 08 2022 08:46:10

%S 96,137712,198579888,286352060064,412919472031680,595429592317621776,

%T 858609059202538568592,1238113667940468298287168,

%U 1785359050561096083591526944,2574486512795432612070683565360,3712407766091963265509842109721456

%N Expansion of x * (96 - 816*x) / ((1 - x) * (1 - 1442*x + x^2)) in powers of x.

%C The continued fraction convergents of sqrt(10) are 3/1, 19/6, 117/37, 721/228, ...

%H Colin Barker, <a href="/A253442/b253442.txt">Table of n, a(n) for n = 1..300</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1443,-1443,1).

%F G.f.: x * (96 - 816*x) / ((1 - x) * (1 - 1442*x + x^2)).

%F a(n) = A253410(2*n) for all n in Z.

%F 1 - a(-n) = A253410(2*n + 1) for all n in Z.

%F From _Colin Barker_, Nov 24 2017: (Start)

%F a(n) = (1/2 - (5+2*sqrt(10))/20*(721+228*sqrt(10))^(-n) + (-1/4 + 1/sqrt(10))*(721+228*sqrt(10))^n).

%F a(n) = 1443*a(n-1) - 1443*a(n-2) + a(n-3) for n>3.

%F (End)

%e G.f. = 96*x + 137712*x^2 + 198579888*x^3 + 286352060064*x^4 + ...

%t CoefficientList[Series[48*x*(2-17*x)/((1-x)*(1-1442*x+x^2)), {x,0,30}], x] (* _G. C. Greubel_, Aug 03 2018 *)

%t LinearRecurrence[{1443,-1443,1},{96,137712,198579888},20] (* _Harvey P. Dale_, Aug 23 2020 *)

%o (PARI) {a(n) = my(t=(721 - 228*quadgen(40))^n); (1 - real(t) - 4*imag(t)) / 2};

%o (PARI) Vec(48*x*(2 - 17*x) / ((1 - x)*(1 - 1442*x + x^2)) + O(x^20)) \\ _Colin Barker_, Nov 24 2017

%o (Magma) m:=20; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(48*x*(2 - 17*x)/((1 - x)*(1 - 1442*x + x^2)))); // _G. C. Greubel_, Aug 03 2018

%Y Cf. A253410.

%K nonn,easy

%O 1,1

%A _Michael Somos_, Dec 31 2014