A005056 a(n) = 3^n + 2^n - 1.

1, 4, 12, 34, 96, 274, 792, 2314, 6816, 20194, 60072, 179194, 535536, 1602514, 4799352, 14381674, 43112256, 129271234, 387682632, 1162785754, 3487832976, 10462450354, 31385253912, 94151567434, 282446313696, 847322163874, 2541932937192, 7625731702714

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Binomial transform of A083313. - Paul Barry, Apr 25 2003

Let P(A) be the power set of an n-element set A and R be a relation on P(A) such that for all x, y of P(A), xRy if either 0) x is a proper subset of y or y is a proper subset of x and x and y are disjoint, 1) x is not a subset of y and y is not a subset of x and x and y are disjoint, or 2) x equals y. Then a(n) = |R|. - Ross La Haye, Mar 19 2009

Links

• Vincenzo Librandi, 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.
• Index entries for linear recurrences with constant coefficients, signature (6,-11,6).

Crossrefs

Cf. A083313.

Programs

Mathematica

Table[3^n + 2^n - 1, {n, 0, 60}] (* Vladimir Joseph Stephan Orlovsky, Jun 27 2011 *)
CoefficientList[Series[(1 - 2 x - x^2) / ((1 - x) (1 - 2 x) (1 - 3 x)), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 08 2013 *)
LinearRecurrence[{6,-11,6},{1,4,12},30] (* Harvey P. Dale, Aug 18 2023 *)

PARI

a(n) = 3^n + 2^n - 1 \\ Rick L. Shepherd, Aug 07 2017

Formula

From Paul Barry, Apr 25 2003: (Start)

G.f.: (1-2x-x^2)/((1-x)(1-2x)(1-3x)).

E.g.f.: exp(3x) + exp(2x) - exp(x). (End)

a(n) = 5*a(n-1) - 6*a(n-2) - 2 for n > 1, a(0)=1, a(1)=4. - Vincenzo Librandi, Dec 31 2010

a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) for n > 2, a(0)=1, a(1)=4, a(2)=12. - Rick L. Shepherd, Aug 07 2017

a(n) = A007689(n)-1. - R. J. Mathar, Mar 10 2022