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a(n) = (7*9^n + 1)/8.
7

%I #52 Oct 19 2024 08:33:36

%S 1,8,71,638,5741,51668,465011,4185098,37665881,338992928,3050936351,

%T 27458427158,247125844421,2224132599788,20017193398091,

%U 180154740582818,1621392665245361,14592533987208248,131332805884874231,1181995252963868078,10637957276674812701,95741615490073314308,861674539410659828771

%N a(n) = (7*9^n + 1)/8.

%C Case r=9 in a(n)=((r-2)*r^n+1)/(r-1).

%H G. C. Greubel, <a href="/A187709/b187709.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 (10,-9).

%F a(n) = (7*9^n + 1)/8.

%F a(n) = +10*a(n-1) -9*a(n-2).

%F a(n) = 8*Sum_{i=0..n-1} a(i) -n + 1.

%F G.f.: (1-2*x)/((1-x)*(1-9*x)).

%F a(n) = 9^n - Sum_{i=0..n-1} 9^i for n>0. - _Bruno Berselli_, Jun 20 2013

%F E.g.f.: (7*exp(9*x) + exp(x))/8. - _G. C. Greubel_, Nov 06 2018

%t (7*9^Range[0,30]+1)/8 (* or *) LinearRecurrence[{10,-9},{1,8},30] (* _Harvey P. Dale_, Jul 20 2012 *)

%o (Magma) [(7*9^n+1)/8: n in [0..25]]; // _Vincenzo Librandi_, Mar 30 2011

%o (PARI) a(n)=(7*9^n+1)/8 \\ _Charles R Greathouse IV_, Oct 07 2015

%Y Cf. A007051, A083068, A059482: cases r=3,8,10 in ((r-2)*r^n+1)/(r-1), respectively.

%K nonn,easy

%O 0,2

%A _Sture Sjöstedt_, Mar 30 2011

%E Additional formulas from _Bruno Berselli_