A248225 a(n) = 6^n - 3^n.

0, 3, 27, 189, 1215, 7533, 45927, 277749, 1673055, 10058013, 60407127, 362619909, 2176250895, 13059099693, 78359381127, 470170635669, 2821066860735, 16926530304573, 101559569247927, 609358577749029, 3656154953278575, 21936940180024653

Offset: 0

Author

Vincenzo Librandi, Oct 04 2014

Keywords

• nonn
• easy

Links

• Index entries for linear recurrences with constant coefficients, signature (9,-18).

Crossrefs

Cf. sequences of the form k^n-3^n: A005061 (k=4), A005058 (k=5), this sequence (k=6), A190541 (k=7), A190543 (k=8), A059410 (k=9), A248226 (k=10), A139741 (k=11).

Cf. A000225, A000244.

Programs

Magma

[6^n-3^n: n in [0..30]];

Mathematica

Table[6^n - 3^n, {n, 0, 25}] (* or *) CoefficientList[Series[3 x / ((1 - 3 x) (1 - 6 x)), {x, 0, 30}], x]
LinearRecurrence[{9,-18},{0,3},30] (* Harvey P. Dale, Jul 12 2025 *)

Formula

G.f.: 3*x/((1-3*x)*(1-6*x)).

a(n) = 9*a(n-1) - 18*a(n-2).

a(n) = 3^n*(2^n - 1) = A000244(n)*A000225(n).

E.g.f.: 2*exp(9*x/2)*sinh(3*x/2). - Elmo R. Oliveira, Mar 31 2025