A092898 Expansion of (1 - 4*x + 4*x^2 - 4*x^3)/(1 - 4*x).

1, 0, 4, 12, 48, 192, 768, 3072, 12288, 49152, 196608, 786432, 3145728, 12582912, 50331648, 201326592, 805306368, 3221225472, 12884901888, 51539607552, 206158430208, 824633720832, 3298534883328, 13194139533312, 52776558133248

Offset

0,3

Comments

Partial sums are A092896.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (4).

Formula

a(n+2) = 4 * A002001(n).

a(n) = (3*4^n + 13*0^n)/16 + Sum_{k=0..n} binomial(n, k)*(-1)^k*(3*k/4 + k*(k-1)/2).

G.f.: 1 - x + 8*x^2 + 2*x/G(0), where G(k) = 1 + 1/(1 - x*(3*k+4)/(x*(3*k+7) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 11 2013

a(n) = A110594(n-1) for n >= 2. - Georg Fischer, Nov 03 2018

From G. C. Greubel, Feb 21 2021: (Start)

a(n) = (3*4^n +16*[n=2] -12*[n=1] +13*0^n)/16.

E.g.f.: (13 -12*x + 8*x^2 + 3*exp(4*x))/16. (End)

Maple

a:= n-> 3*4^n/16+13*0^n/16+add(binomial(n, k)*(-1)^k*(3*k/4+k*(k-1)/2), k=0..n):
seq(a(n), n=0..30);  # Alois P. Heinz, Nov 03 2018

Mathematica

Join[{1, 0, 4}, LinearRecurrence[{4}, {12}, 22]] (* Jean-François Alcover, Sep 16 2019 *)

Prog

(PARI) Vec((1 -4*x +4*x^2 -4*x^3)/(1-4*x) + O(x^30)) \\ Andrew Howroyd, Nov 03 2018
(Sage) [1, 0, 4]+[3*4^(n-2) for n in (3..30)] # G. C. Greubel, Feb 21 2021
(Magma) [1, 0, 4] cat [3*4^(n-2): n in [3..30]]; // G. C. Greubel, Feb 21 2021

Crossrefs

Cf. A002001, A092896, A110594.

Sequence in context: A019309 A056632 A149385 * A110594 A151483 A355800

Adjacent sequences: A092895 A092896 A092897 * A092899 A092900 A092901

Keyword

easy,nonn

Author

Paul Barry, Mar 12 2004

Status

approved