A193410 Expansion of (1-3*x)/(1-6*x+18*x^2).

1, 3, 0, -54, -324, -972, 0, 17496, 104976, 314928, 0, -5668704, -34012224, -102036672, 0, 1836660096, 11019960576, 33059881728, 0, -595077871104, -3570467226624, -10711401679872, 0, 192805230237696, 1156831381426176, 3470494144278528

Offset: 0

Author

Bruno Berselli, Aug 04 2011

Keywords

• sign
• easy

Comments

Also real parts of 3^n*(1+i)^n, where i=sqrt(-1).

If |a(n)| > 0 then it is in A130505.

Links

• Bruno Berselli, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (6,-18).

Crossrefs

Cf. A066771, A121622, A130505.

Programs

Magma

m:=26; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-3*x)/(1-6*x+18*x^2))); /* or */ &cat[[r,3*r,0,-54*r] where r is (-324)^n: n in [0..6]];

Magma

I:=[1, 3]; [n le 2 select I[n] else 6*Self(n-1)-18*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Mar 26 2013

Mathematica

CoefficientList[Series[(1 - 3 x)/(1 - 6 x + 18 x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Mar 26 2013 *)
LinearRecurrence[{6,-18},{1,3},40] (* Harvey P. Dale, Jul 27 2021 *)

Maxima

makelist(coeff(taylor((1-3*x)/(1-6*x+18*x^2), x, 0, n), x, n), n, 0, 25);

PARI

Vec((1-3*x)/(1-6*x+18*x^2) +O(x^26))

Formula

G.f.: (1-3*x)/(1-6*x+18*x^2).

a(n) = 3^n*A146559(n) = (1/2)*((3+3*i)^n+(3-3*i)^n), where i=sqrt(-1).

a(n) = 6*a(n-1)-18*a(n-2) for n>1.

a(n) = (3*sqrt(2))^n*cos(pi*n/4).

a(4k+2) = 0, a(4k+1) = 3*a(4k) = 18*a(4k-1) = 3*(-324)^k.

G.f.: W(0)/2, where W(k) = 1 + 1/(1 - x*(3*k+3)/(x*(3*k+6) + 1/W(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 28 2013