A091344 a(n) = 2*3^n - 3*2^n + 1.

0, 1, 7, 31, 115, 391, 1267, 3991, 12355, 37831, 115027, 348151, 1050595, 3164071, 9516787, 28599511, 85896835, 257887111, 774054547, 2322950071, 6970423075, 20914414951, 62749536307, 188261191831, 564808741315, 1694476555591

Offset: 0

Author

Mario Catalani (mario.catalani(AT)unito.it), Jan 01 2004

Keywords

• easy
• nonn

Comments

Starting with offset 1 = binomial transform of A068293: (1, 6, 18, 42, 90, ...) and double binomial transform of (1, 5, 7, 5, 7, 5, ...). - Gary W. Adamson, Jan 13 2009

Number of pairs (A,B) where A and B are nonempty subsets of {1,2,...,n} and one of these subsets is a subset of the other. - For the case that one of these subsets is a proper subset of the other see a(n+1) in A260217. - If empty subsets are included, see A027649 (all subsets) and A056182 (proper subsets). - Manfred Boergens, Aug 02 2023

Links

• Christian Ballot and Florian Luca, Prime factors of a^f(n)-1 with an irreducible polynomial f(x),New York J. Math. 12 (2006), 39-45 (electronic).
• Christian Ballot and Florian Luca, Common prime factors of a^n-b and c^n-d, Unif. Distrib. Theory 2 (2007), no. 2, 19-34 (electronic).
• John Elias, Illustration of initial terms: Sixfold Sierpinski Stars
• Index entries for linear recurrences with constant coefficients, signature (6,-11,6).

Crossrefs

Cf. A027649, A056182, A068293, A260217.

Programs

Maple

a:=n->sum((3^(n-j-1)-2^(n-2-j))*12, j=0..n): seq(a(n), n=-1..24); # Zerinvary Lajos, Feb 11 2007
with (combinat):a:=n->stirling2(n,3)+stirling2(n+1,3): seq(a(n), n=1..26); # Zerinvary Lajos, Oct 07 2007

Mathematica

Table[Sum[i!i^2 StirlingS2[n, i](-1)^(n - i), {i, 1, n}], {n, 0, 30}]
Table[2*3^n-3*2^n+1,{n,0,30}] (* or *) LinearRecurrence[{6,-11,6},{0,1,7},30] (* Harvey P. Dale, Dec 31 2013 *)

Formula

a(n) = Sum_{i=1..n} i!*i^2*Stirling2(n,i)*(-1)^(n-i).

From Christian Ballot via R. K. Guy, Jan 13 2009: (Start)

a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3);

G.f.: x*(1+x)/((1-x)*(2-x)*(3-x)). (End)

a(n) = 5*a(n-1) - 6*a(n-2) + 2, a(0)=0, a(1)=1. - Vincenzo Librandi, Nov 25 2010

E.g.f.: exp(x)*(1 - 3*exp(x) + 2*exp(2*x)). - Stefano Spezia, May 18 2024

Extensions

Edited by N. J. A. Sloane, Jan 13 2009 at the suggestion of R. K. Guy; the concise definition was provided by Vladeta Jovovic, Jan 01 2004