A090400 Expansion of 1/(1-3x+3x^3) in powers of x.

1, 3, 9, 24, 63, 162, 414, 1053, 2673, 6777, 17172, 43497, 110160, 278964, 706401, 1788723, 4529277, 11468628, 29039715, 73531314, 186188058, 471445029, 1193741145, 3022659261, 7653642696, 19379704653, 49071136176, 124252480440

Offset

0,2

Links

Table of n, a(n) for n=0..27.

Index entries for linear recurrences with constant coefficients, signature (3,0,-3).

Formula

a(n)=3a(n-1)-3a(n-3); a(n)=sum{k=0..floor(n/2), comb(n-2k, k) (-1)^k 3^(n-2k) }.

Example

G.f. = 1 + 3*x + 9*x^2 + 24*x^3 + 63*x^4 + 162*x^5 + 414*x^6 + 1053*x^7 + ...

Mathematica

LinearRecurrence[{3, 0, -3}, {1, 3, 9}, 30] (* Harvey P. Dale, Jan 08 2021 *)

Prog

(PARI) {a(n) = sum(k=0, n\3, binomial(n - 2*k, k) * (-1)^k * 3^(n - 2*k))}; /* Michael Somos, Jan 30 2015 */
(PARI) {a(n) = if( n<0, 0, polcoeff( 1 / (1 - 3*x + 3*x^3) + x * O(x^n), n))}; /* Michael Somos, Jan 30 2015 */

Crossrefs

Sequence in context: A098690 A267960 A268938 * A123888 A166290 A097134

Adjacent sequences: A090397 A090398 A090399 * A090401 A090402 A090403

Keyword

easy,nonn

Author

Paul Barry, Nov 28 2003

Status

approved