A094374 a(n) = (3^n-1)/2 + 2^n.

1, 3, 8, 21, 56, 153, 428, 1221, 3536, 10353, 30548, 90621, 269816, 805353, 2407868, 7207221, 21588896, 64701153, 193972388, 581655021, 1744440776, 5232273753, 15694724108, 47079978021, 141231545456, 423677859153, 1271000023028, 3812932960221, 11438664662936

Offset: 0

Author

Paul Barry, Apr 28 2004

Keywords

• easy
• nonn

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).

Crossrefs

Cf. A000079, A000225, A000392, A003462, A094373, A125103.

Programs

Magma

[(3^n-1)/2+2^n: n in [0..30]]; // Vincenzo Librandi, Nov 30 2015

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 *)

PARI

a(n)=(3^n-1)/2+2^n \\ Charles R Greathouse IV, Oct 16 2015

SageMath

[(3^n +2^(n+1) -1)//2 for n in range(31)] # G. C. Greubel, Sep 26 2024

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) = A003462(n) + A000079(n).

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