OFFSET
0,2
COMMENTS
Binomial transform of A094373.
Row sums of A125103. - Paul Barry, Dec 04 2007
Let P(A) be the power set of an n-element set A. Then a(n) = the number of pairs of elements {x,y} of P(A) for which either 0) x and y are disjoint and for which either x is a subset of y or y is a subset of x, or 1) x and y are disjoint and for which x is not a subset of y and y is not a subset of x, or 2) x = y. - Ross La Haye, Jan 11 2008
a(n) is the number of words of length n over the alphabet {0,1,2} with an even number of occurrences of the substring 01. - Daimon S. Mayorga, Sep 10 2020
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
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.
A. Prasad, Equivalence classes of nodes in trees and rational generating functions, arXiv preprint arXiv:1407.5284 [math.CO], 2014.
Index entries for linear recurrences with constant coefficients, signature (6,-11,6).
FORMULA
G.f.: (1-3x+x^2)/((1-x)*(1-2x)*(1-3x)).
a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3).
a(n) = Sum_{k=0..n} C(n,k)+2^k*C(n,k+1). - Paul Barry, Dec 04 2007
a(n) = StirlingS2(n+1,3) + 2*StirlingS2(n+1,2) + 1. - Ross La Haye, Jan 11 2008
E.g.f.: exp(2*x)*(1 + sinh(x)). - G. C. Greubel, Sep 26 2024
MATHEMATICA
Table[(3^n-1)/2+2^n, {n, 0, 30}] (* or *) LinearRecurrence[{6, -11, 6}, {1, 3, 8}, 30] (* Harvey P. Dale, Jul 22 2013 *)
PROG
(PARI) a(n)=(3^n-1)/2+2^n \\ Charles R Greathouse IV, Oct 16 2015
(Magma) [(3^n-1)/2+2^n: n in [0..30]]; // Vincenzo Librandi, Nov 30 2015
(SageMath) [(3^n +2^(n+1) -1)//2 for n in range(31)] # G. C. Greubel, Sep 26 2024
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Apr 28 2004
STATUS
approved