A001117 a(n) = 3^n - 3*2^n + 3.

1, 0, 0, 6, 36, 150, 540, 1806, 5796, 18150, 55980, 171006, 519156, 1569750, 4733820, 14250606, 42850116, 128746950, 386634060, 1160688606, 3483638676, 10454061750, 31368476700, 94118013006, 282379204836, 847187946150, 2541664501740, 7625194831806

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Differences of 0. Labeled ordered partitions into 3 parts.

Number of surjections from an n-element set onto a three-element set, with n >= 3. - Mohamed Bouhamida, Dec 15 2007

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 not a subset of y and y is not a subset of x and x and y are disjoint, or 1) x is a proper subset of y or y is a proper subset of x and x and y are intersecting. Then a(n+1) = |R|. - Ross La Haye, Mar 19 2009

For n>0, the number of rows of n colors using exactly three colors. For n=3, the six rows are ABC, ACB, BAC, BCA, CAB, and CBA. - Robert A. Russell, Sep 25 2018

References

• H. T. Davis, Tables of the Mathematical Functions. Vols. 1 and 2, 2nd ed., 1963, Vol. 3 (with V. J. Fisher), 1962; Principia Press of Trinity Univ., San Antonio, TX, Vol. 2, p. 212.
• J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 33.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
• J. F. Steffensen, Interpolation, 2nd ed., Chelsea, NY, 1950, see p. 54.
• A. H. Voigt, Theorie der Zahlenreihen und der Reihengleichungen, Goschen, Leipzig, 1911, p. 31.

Links

• T. D. Noe, Table of n, a(n) for n = 0..200
• John Elias, Illustration of initial terms: Interior Sierpinski triangle
• K. S. Immink, Coding Schemes for Multi-Level Channels that are Intrinsically Resistant Against Unknown Gain and/or Offset Using Reference Symbols, 2013.
• 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.
• P. A. Piza, Kummer numbers, Mathematics Magazine, 21 (1947/1948), 257-260.
• P. A. Piza, Kummer numbers, Mathematics Magazine, 21 (1947/1948), 257-260. [Annotated scanned copy]
• Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
• A. H. Voigt, Theorie der Zahlenreihen und der Reihengleichungen, Leipzig, 1911.
• A. H. Voigt, Theorie der Zahlenreihen und der Reihengleichungen, Goschen, Leipzig, 1911. [Annotated scans of pages 30-33 only]
• Index entries for linear recurrences with constant coefficients, signature (6,-11,6).

Crossrefs

Cf. A000919, A001118.

Column 3 of A019538 (n>0).

Programs

Maple

with(combstruct):ZL:=[S,{S=Sequence(U,card=r),U=Set(Z,card>=1)}, labeled]: 1, seq(count(subs(r=3,ZL),size=m),m=1..25); # Zerinvary Lajos, Mar 09 2007
A001117:=-6/(z-1)/(3*z-1)/(2*z-1); # Conjectured by Simon Plouffe in his 1992 dissertation. Gives sequence except for three leading terms.

Mathematica

k=3; Prepend[Table[k!StirlingS2[n,k],{n,1,30}],1] (* Robert A. Russell, Sep 25 2018 *)

PARI

a(n)=3^n-3*2^n+3 \\ Charles R Greathouse IV, Sep 24 2015

Formula

a(n) = [n=0] + 3!*S(n, 3).

E.g.f.: 1 + (exp(x)-1)^3.

For n>=3: a(n+1) = 3*a(n) + 3*(2^n - 2) = 3*a(n) + 3*A000918(n). - Geoffrey Critzer, Feb 27 2009

G.f.: (-1-11*x^2+6*x)/((x-1)*(3*x-1)*(2*x-1)). - Maksym Voznyy (voznyy(AT)mail.ru), Jul 26 2009

Extensions

Extended with formula and alternate description by Christian G. Bower, Aug 15 1998

Simpler description from Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 07 2001