A298271 - OEIS

%I #37 Dec 03 2021 00:35:30

%S 0,1,323,104006,33489610,10783550415,3472269744021,1118060074024348,

%T 360011871566096036,115922704584208899245,37326750864243699460855,

%U 12019097855581887017496066,3870112182746503375934272398,1246164103746518505163818216091

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

%H Colin Barker, <a href="/A298271/b298271.txt">Table of n, a(n) for n = 0..399</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (323,-323,1).

%F G.f.: x/((1 - x)*(1 - 322*x + x^2)).

%F a(n) = a(-n-1) = 323*a(n-1) - 323*a(n-2) + a(n-3).

%F a(n) = (1/5760)*((2 + sqrt(5))^(4*n+2) + (2 + sqrt(5))^-(4*n+2) - 18).

%F a(n) = A298101(n) - A298101(n-1) + A298101(n-2) - A298101(n-3) + ..., hence:

%F a(n) + a(n-1) = A298101(n).

%F a(n) - a(n-1) = (1/144)*Fibonacci(12*n).

%F a(n) - a(n-2) = (1/8)*Fibonacci(12*n-6).

%F a(n)*a(n-2) = a(n-1)*(a(n-1) - 1).

%F Sum_{j>1} 1/a(j) = 161 - 72*sqrt(5) = A094214^12.

%F a(n) = A157459(n+1)/72. - _Greg Dresden_, Dec 02 2021

%t CoefficientList[x/((1 - x) (1 - 322 x + x^2)) + O[x]^20, x]

%o (Sage)

%o gf = x/((1-x)*(1-322*x+x^2))

%o print(taylor(gf, x, 0, 20).list())

%o (Maxima) makelist(coeff(taylor(x/((1-x)*(1-322*x+x^2)), x, 0, n), x, n), n, 0, 20);

%o (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 (PARI) concat(0, Vec(x / ((1 - x)*(1 - 322*x + x^2)) + O(x^15))) \\ _Colin Barker_, Jan 19 2018

%Y Cf. A000045, A253368, A298101.

%K nonn,easy

%O 0,3

%A _Bruno Berselli_, Jan 16 2018