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A248216 a(n) = 6^n - 2^n. 4

%I #22 Mar 27 2022 13:39:32

%S 0,4,32,208,1280,7744,46592,279808,1679360,10077184,60465152,

%T 362795008,2176778240,13060685824,78364147712,470184951808,

%U 2821109841920,16926659313664,101559956406272,609359739486208,3656158439014400,21936950638280704

%N a(n) = 6^n - 2^n.

%H G. C. Greubel, <a href="/A248216/b248216.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (8,-12).

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

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

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

%F E.g.f.: exp(6*x) - exp(2*x) = 2*exp(4*x)*sinh(2*x). - _G. C. Greubel_, Feb 09 2021

%F a(n) = 4*A016129(n-1). - _R. J. Mathar_, Mar 10 2022

%F a(n) = A000400(n) - A000079(n). - _Bernard Schott_, Mar 27 2022

%t Table[6^n - 2^n, {n, 0, 25}] (* or *) CoefficientList[Series[4x/((1-2x)(1-6x)), {x, 0, 30}], x]

%t LinearRecurrence[{8,-12},{0,4},30] (* _Harvey P. Dale_, Dec 21 2019 *)

%o (Magma) [6^n-2^n: n in [0..25]];

%o (Sage) [2^n*(3^n -1) for n in (0..25)] # _G. C. Greubel_, Feb 09 2021

%Y 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).

%Y Cf. A000079, A000400, A024023.

%K nonn,easy

%O 0,2

%A _Vincenzo Librandi_, Oct 04 2014

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Last modified March 29 00:26 EDT 2024. Contains 371264 sequences. (Running on oeis4.)