A016142 Expansion of 1/((1-3*x)*(1-9*x)).

1, 12, 117, 1080, 9801, 88452, 796797, 7173360, 64566801, 581120892, 5230147077, 47071500840, 423644039001, 3812797945332, 34315186290957, 308836690967520, 2779530261754401, 25015772484929772, 225141952751788437, 2026277575928357400, 18236498186842001001, 164128483692038362212

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• changed

Comments

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

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• A. M. Hinz, S. Klavžar, U. Milutinović, C. Petr, The Tower of Hanoi - Myths and Maths, Birkhäuser 2013. See page 100. Book's website.
• Index entries for linear recurrences with constant coefficients, signature (12,-27).

Crossrefs

Cf. A006100, A090245.

Programs

Magma

[(1/6)*(9^(n+1)-3^(n+1)): n in [0..20]]; // Vincenzo Librandi, Feb 24 2014

Mathematica

Join[{a=1,b=12},Table[c=12*b-27*a;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 31 2011 *)
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 *)
Table[ncards = 3^nattr; (ncards*(ncards - 1))/6, {nattr, 1, 20}] (* Robert Price, Oct 14 2017 *)

PARI

Vec(1/((1-3*x)*(1-9*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

PARI

a(n) = (1/6)*(9^(n+1) - 3^(n+1)); \\ Joerg Arndt, Feb 23 2014

Sage

[lucas_number1(n,12,27) for n in range(1, 20)] # Zerinvary Lajos, Apr 27 2009

Formula

a(n) = (1/6)*(9^(n+1) - 3^(n+1)).

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

a(n) = 12*a(n-1) - 27*a(n-2), a(0)=1, a(1)=12. - Vincenzo Librandi, Mar 14 2011

a(n) = A006100(n+2) - A006100(n+1), for n > 0. - Álvar Ibeas, Nov 29 2015

E.g.f.: exp(3*x)*(3*exp(3*x) - 1)/2. - Elmo R. Oliveira, Mar 08 2025