A248216 a(n) = 6^n - 2^n.

0, 4, 32, 208, 1280, 7744, 46592, 279808, 1679360, 10077184, 60465152, 362795008, 2176778240, 13060685824, 78364147712, 470184951808, 2821109841920, 16926659313664, 101559956406272, 609359739486208, 3656158439014400, 21936950638280704

Offset: 0

Author

Vincenzo Librandi, Oct 04 2014

Keywords

• nonn
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (8,-12).

Crossrefs

Sequences of the form k^n - 2^n: A001047 (k=3), A020522 (k=4), A005057 (k=5), this sequence (k=6), A190540 (k=7), A248217 (k=8), A191465 (k=9), A060458 (k=10), A139740 (k=11).

Cf. A000079, A000400, A024023.

Programs

Magma

[6^n-2^n: n in [0..25]];

Mathematica

Table[6^n - 2^n, {n, 0, 25}] (* or *) CoefficientList[Series[4x/((1-2x)(1-6x)), {x, 0, 30}], x]
LinearRecurrence[{8,-12},{0,4},30] (* Harvey P. Dale, Dec 21 2019 *)

Sage

[2^n*(3^n -1) for n in (0..25)] # G. C. Greubel, Feb 09 2021

Formula

G.f.: 4*x/((1-2*x)*(1-6*x)).

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

a(n) = 2^n*(3^n - 1) = A000079(n) * A024023(n).

E.g.f.: exp(6*x) - exp(2*x) = 2*exp(4*x)*sinh(2*x). - G. C. Greubel, Feb 09 2021

a(n) = 4*A016129(n-1). - R. J. Mathar, Mar 10 2022

a(n) = A000400(n) - A000079(n). - Bernard Schott, Mar 27 2022