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%I #60 Sep 28 2023 14:26:35
%S 1,12,117,1080,9801,88452,796797,7173360,64566801,581120892,
%T 5230147077,47071500840,423644039001,3812797945332,34315186290957,
%U 308836690967520,2779530261754401,25015772484929772,225141952751788437
%N Expansion of 1/((1-3x)(1-9x)).
%C a(n) is the number of lattices L in Z^(n+1) such that the quotient group Z^(n+1) / L is C_9. - _Álvar Ibeas_, Nov 29 2015
%C In the game of SET with four attributes there are 1080 potential SETs. See A090245. In the generalized game of SET with different numbers of attributes, the number of potential SETs is a(n+1). - _Robert Price_, Oct 14 2017
%H Vincenzo Librandi, <a href="/A016142/b016142.txt">Table of n, a(n) for n = 0..200</a>
%H A. M. Hinz, S. Klavžar, U. Milutinović, C. Petr, <a href="http://dx.doi.org/10.1007/978-3-0348-0237-6">The Tower of Hanoi - Myths and Maths</a>, Birkhäuser 2013. See page 100. <a href="http://tohbook.info">Book's website</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (12, -27).
%F a(n) = (1/6)*(9^(n+1) - 3^(n+1)).
%F a(n-1) = Sum_{i=1..n} binomial(n, i)*3^(n-i)*6^(i-1). - Sam Handler (sam_5_5_5_0(AT)yahoo.com), Jun 29 2004
%F a(n) = 12*a(n-1) - 27*a(n-2), a(0)=1, a(1)=12. - _Vincenzo Librandi_, Mar 14 2011
%F a(n) = A006100(n+2) - A006100(n+1), for n > 0. - _Álvar Ibeas_, Nov 29 2015
%t Join[{a=1,b=12},Table[c=12*b-27*a;a=b;b=c,{n,60}]] (* _Vladimir Joseph Stephan Orlovsky_, Jan 31 2011 *)
%t CoefficientList[Series[1/((1-3x)(1-9x)),{x,0,20}],x] (* or *) Table[ (9^(n+1) -3^(n+1))/6,{n,0,20}] (* _Harvey P. Dale_, Apr 03 2011 *)
%t Table[ncards = 3^nattr; (ncards*(ncards - 1))/6, {nattr, 1, 20}] (* _Robert Price_, Oct 14 2017 *)
%o (Sage) [lucas_number1(n,12,27) for n in range(1, 20)] # _Zerinvary Lajos_, Apr 27 2009
%o (PARI) Vec(1/((1-3*x)*(1-9*x))+O(x^99)) \\ _Charles R Greathouse IV_, Sep 26 2012
%o (PARI) a(n) = (1/6)*(9^(n+1) - 3^(n+1)); \\ _Joerg Arndt_, Feb 23 2014
%o (Magma) [(1/6)*(9^(n+1)-3^(n+1)): n in [0..20]]; // _Vincenzo Librandi_, Feb 24 2014
%Y Cf. A006100, A090245.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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