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A298271 Expansion of x/((1 - x)*(1 - 322*x + x^2)).

%I A298271 #37 Dec 03 2021 00:35:30
%S A298271 0,1,323,104006,33489610,10783550415,3472269744021,1118060074024348,
%T A298271 360011871566096036,115922704584208899245,37326750864243699460855,
%U A298271 12019097855581887017496066,3870112182746503375934272398,1246164103746518505163818216091
%N A298271 Expansion of x/((1 - x)*(1 - 322*x + x^2)).
%H A298271 Colin Barker, <a href="/A298271/b298271.txt">Table of n, a(n) for n = 0..399</a>
%H A298271 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (323,-323,1).
%F A298271 G.f.: x/((1 - x)*(1 - 322*x + x^2)).
%F A298271 a(n) = a(-n-1) = 323*a(n-1) - 323*a(n-2) + a(n-3).
%F A298271 a(n) = (1/5760)*((2 + sqrt(5))^(4*n+2) + (2 + sqrt(5))^-(4*n+2) - 18).
%F A298271 a(n) = A298101(n) - A298101(n-1) + A298101(n-2) - A298101(n-3) + ..., hence:
%F A298271 a(n) + a(n-1) = A298101(n).
%F A298271 a(n) - a(n-1) = (1/144)*Fibonacci(12*n).
%F A298271 a(n) - a(n-2) = (1/8)*Fibonacci(12*n-6).
%F A298271 a(n)*a(n-2) = a(n-1)*(a(n-1) - 1).
%F A298271 Sum_{j>1} 1/a(j) = 161 - 72*sqrt(5) = A094214^12.
%F A298271 a(n) = A157459(n+1)/72. - _Greg Dresden_, Dec 02 2021
%t A298271 CoefficientList[x/((1 - x) (1 - 322 x + x^2)) + O[x]^20, x]
%o A298271 (Sage)
%o A298271 gf = x/((1-x)*(1-322*x+x^2))
%o A298271 print(taylor(gf, x, 0, 20).list())
%o A298271 (Maxima) makelist(coeff(taylor(x/((1-x)*(1-322*x+x^2)), x, 0, n), x, n), n, 0, 20);
%o A298271 (PARI) a(n)=([0,1,0; 0,0,1; 1,-323,323]^n*[0;1;323])[1,1] \\ _Charles R Greathouse IV_, Jan 18 2018
%o A298271 (PARI) concat(0, Vec(x / ((1 - x)*(1 - 322*x + x^2)) + O(x^15))) \\ _Colin Barker_, Jan 19 2018
%Y A298271 Cf. A000045, A253368, A298101.
%K A298271 nonn,easy
%O A298271 0,3
%A A298271 _Bruno Berselli_, Jan 16 2018