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

0, 1, 323, 104006, 33489610, 10783550415, 3472269744021, 1118060074024348, 360011871566096036, 115922704584208899245, 37326750864243699460855, 12019097855581887017496066, 3870112182746503375934272398, 1246164103746518505163818216091

Offset: 0

Author

Bruno Berselli, Jan 16 2018

Keywords

• nonn
• easy

Links

• Colin Barker, Table of n, a(n) for n = 0..399
• Index entries for linear recurrences with constant coefficients, signature (323,-323,1).

Crossrefs

Cf. A000045, A253368, A298101.

Programs

Mathematica

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

Maxima

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

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

PARI

concat(0, Vec(x / ((1 - x)*(1 - 322*x + x^2)) + O(x^15))) \\ Colin Barker, Jan 19 2018

Sage

gf = x/((1-x)*(1-322*x+x^2))
print(taylor(gf, x, 0, 20).list())

Formula

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

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

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

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

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

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

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

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

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

a(n) = A157459(n+1)/72. - Greg Dresden, Dec 02 2021