OFFSET
0,4
COMMENTS
Binomial transform of A000295.
a(n) = A112626(n, 2). - Ross La Haye, Jan 11 2006
Let Q be a binary relation on the power set P(A) of a set A having n = |A| elements such that for all x,y of P(A), xQy if x is a proper subset of y and |y| - |x| > 1. Then a(n) = |Q|. - Ross La Haye, Jan 11 2008
a(n) is the number of n-digit ternary sequences that have at least two 0's. - Geoffrey Critzer, Apr 14 2009
LINKS
Harry J. Smith, Table of n, a(n) for n = 0..200
Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
Index entries for linear recurrences with constant coefficients, signature (7,-16,12).
FORMULA
a(n) = 3^n - 2^n - n*2^(n-1).
From Ross La Haye, Apr 26 2006: (Start)
a(n) = Sum_{k=2..n} binomial(n,k)2^(n-k). (End)
Inverse binomial transform of A086443. - Ross La Haye, Apr 29 2006
From Geoffrey Critzer, Apr 14 2009: (Start)
E.g.f.: exp(2*x)*(exp(x) - x - 1).
a(n) = 3*a(n-1) + (n-1)*2^(n-2). (End)
MAPLE
seq(3^n - 2^n - n*2^(n-1), n=0..30); # G. C. Greubel, Nov 18 2019
MATHEMATICA
RecurrenceTable[{a[n]==3*a[n-1] + (n-1) 2^(n-2), a[0]==0}, a, {n, 0, 30}] (* Geoffrey Critzer, Apr 14 2009 *)
CoefficientList[Series[x^2/((1-3x)(1-2x)^2), {x, 0, 30}], x] (* Vincenzo Librandi, Nov 29 2015 *)
PROG
(PARI) for(n=0, 50, write("b066810.txt", n, " ", 3^n -2^n -n*2^(n-1)) ) \\ Harry J. Smith, Mar 29 2010
(Magma) [3^n-2^n-n*2^(n-1): n in [0..30]]; // Vincenzo Librandi, Nov 29 2015
(Sage) [3^n - 2^n - n*2^(n-1) for n in (0..30)] # G. C. Greubel, Nov 18 2019
(GAP) List([0..30], n-> 3^n - 2^n - n*2^(n-1)); # G. C. Greubel, Nov 18 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Jan 25 2002
EXTENSIONS
Additional comments from Ross La Haye, Sep 27 2005
STATUS
approved