A024102 - OEIS

%I #21 Sep 09 2024 21:06:37

%S 1,8,79,726,6557,59044,531435,4782962,43046713,387420480,3486784391,

%T 31381059598,282429536469,2541865828316,22876792454947,

%U 205891132094634,1853020188851825,16677181699666552,150094635296999103,1350851717672992070,12157665459056928781,109418989131512359188

%N a(n) = 9^n - n.

%H Vincenzo Librandi, <a href="/A024102/b024102.txt">Table of n, a(n) for n = 0..400</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (11,-19,9).

%F From _Vincenzo Librandi_, Jun 17 2013: (Start)

%F G.f.: (1-3*x+10*x^2)/((1-9*x)(1-x)^2).

%F a(n) = 11*a(n-1) - 19*a(n-2) + 9*a(n-3). (End)

%F E.g.f.: exp(x)*(exp(8*x) - x). - _Elmo R. Oliveira_, Sep 09 2024

%t Table[9^n - n, {n, 0, 20}] (* or *) CoefficientList[Series[(1 - 3 x + 10 x^2) / ((1 - 9 x) (1 - x)^2), {x, 0, 30}], x] (* _Vincenzo Librandi_, Jun 17 2013 *)

%o (Magma) [9^n-n: n in [0..25]]; // _Vincenzo Librandi_, Jul 06 2011

%o (Magma) I:=[1, 8, 79]; [n le 3 select I[n] else 11*Self(n-1)-19*Self(n-2)+9*Self(n-3): n in [1..30]]; // _Vincenzo Librandi_, Jun 17 2013

%o (PARI) a(n)=9^n-n \\ _Charles R Greathouse IV_, Oct 07 2015

%Y Cf. numbers of the form k^n - n: A000325 (k=2), A024024 (k=3), A024037 (k=4), A024050 (k=5), A024063 (k=6), A024076 (k=7), A024089 (k=8), this sequence (k=9), A024115 (k=10), A024128 (k=11), A024141 (k=12).

%Y Cf. A198966 (first differences).

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_