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

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

Offset

0,2

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).

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

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 *)

Prog

(Sage) [lucas_number1(n, 12, 27) for n in range(1, 20)] # Zerinvary Lajos, Apr 27 2009
(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
(Magma) [(1/6)*(9^(n+1)-3^(n+1)): n in [0..20]]; // Vincenzo Librandi, Feb 24 2014

Crossrefs

Cf. A006100, A090245.

Sequence in context: A304827 A182671 A154346 * A105218 A180777 A163950

Adjacent sequences: A016139 A016140 A016141 * A016143 A016144 A016145

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved