A000247 a(n) = 2^n - n - 2. (Formerly M2836 N1141)

0, 3, 10, 25, 56, 119, 246, 501, 1012, 2035, 4082, 8177, 16368, 32751, 65518, 131053, 262124, 524267, 1048554, 2097129, 4194280, 8388583, 16777190, 33554405, 67108836, 134217699, 268435426, 536870881, 1073741792, 2147483615

Offset

2,2

Comments

Ways of placing n+1 labeled balls into 2 indistinguishable boxes with at least 2 balls in each box.

2^a(n) is an integer of the form 1/(2 - Sum_{i=1..m} i/2^i). - Benoit Cloitre, Oct 25 2002

Number of permutations avoiding 13-2 that contain the pattern 23-1 exactly twice.

Cost of ternary maximum height Huffman tree with N internal nodes (non-leaves) for minimizing absolutely ordered sequences of size n = 2N + 1. - Alex Vinokur (alexvn(AT)barak-online.net), Nov 02 2004

a(n) is the number of Dyck n-paths whose third upstep initiates the last long ascent, n >= 1. A long ascent is one consisting of 2 or more upsteps. For example, a(3)=3 counts UUDuUDDD, UDUDuUDD, UUDDuUDD (third upstep in small type). - David Callan, Dec 08 2004

Subsequence of A158581; A000120(a(n)) > 1. - Reinhard Zumkeller, Apr 16 2009

Vertices of the tropical Grassmannian simplicial complex G(2,n), related to phylogenetic trees. - Tom Copeland, Oct 03 2011

(Conjecture) Let a(2)=0. For n > 2, let N = 2*n + 1. For each n, define the n X n tridiagonal unit-primitive matrix (see [Jeffery]) A_{N,1}=[0,1,0,...,0; 1,0,1,0,...,0; 0,1,0,1,0,...,0; ...; 0,...,0,1,0,1; 0,...,0,1,1] associated with N. Define the n-dimensional column vectors V_N = [v_1,v_2,...,v_n]^T = [A_{N,1}]^n*[1,1,...,1]^T, where [.]^T denotes matrix transpose and [1,...,1] is the n-dimensional unit vector. Let (v_k)_N denote the k-th element of V_N, k in {1,...,n}. Then a(n) = (v_(n-2))_N. - L. Edson Jeffery, Jan 20 2012

For n>0, (a(n)) is row 3 of the convolution array A213568. - Clark Kimberling, Jun 20 2012

For n>2, a(n-2) is the number of connected induced (non-null) subgraphs of the n-centipede graph. - Giovanni Resta, May 04 2017

References

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 222.

F. N. David and D. E. Barton, Combinatorial Chance. Hafner, NY, 1962, p. 296.

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 76.

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).

Links

T. D. Noe, Table of n, a(n) for n = 2..300

Antal E. Fekete, Apropos Two Notes on Notation, The Amer. Math. Monthly, Vol. 101, No. 8 (Oct., 1994), pp. 771-778. See p. 776.

Robert Israel et al, Primes 2^n - n - 2, Mathematics StackExchange.

L. E. Jeffery, Unit-primitive matrices

T. Mansour, Restricted permutations by patterns of type 2-1, arXiv:math/0202219 [math.CO], 2002.

Mathoverflow, Face numbers for tropical Grassmannian G'_2,7 simplicial complex?

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.

Erik Vigren and Andreas Dieckmann, A New Result in Form of Finite Triple Sums for a Series from Ramanujan's Notebooks, Symmetry (2022) Vol. 14, No. 6, 1090.

Alex Vinokur, Fibonacci-like polynomials produced by m-ary Huffman codes for absolutely ordered sequences, arXiv:cs/0411002 [cs.DM], 2004.

Eric Weisstein's World of Mathematics, Centipede Graph

Eric Weisstein's World of Mathematics, Connected Dominating Set

Eric Weisstein's World of Mathematics, Vertex-Induced Subgraph

Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Formula

E.g.f.: (exp(x)-1-x)*(exp(x)-1).

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

a(n) = 2*a(n-1) + n + 3 = a(n-1) + 2^(n-1) - 1 = A000295(n) - 1 = A000295(n+1) - 2^(n+1).

A107907(a(n)) = A000225(n). - Reinhard Zumkeller, May 28 2005

Starting (3, 10, 25, 56, ...) = binomial transform of [3, 7, 8, 8, 8, ...]. - Gary W. Adamson, Nov 07 2007

a(2)=0, a(3)=3, a(4)=10, a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3). - Harvey P. Dale, Aug 23 2011

a(n) = (Sum_{k=2..floor(n/2)} binomial(n+1,k)) + if(n odd, binomial(n+1,(n+1)/2)/2, 0).

a(n) = Sum_{k=0..n-3} Sum_{i=0..n-1} C(i,k). - Wesley Ivan Hurt, Sep 20 2017

Example

a(3) = 4!/(2!*2!*2!) = 3.

Maple

A000247:=(-3+2*z)/((2*z-1)*(z-1)**2); # Simon Plouffe in his 1992 dissertation

Mathematica

LinearRecurrence[{4, -5, 2}, {0, 3, 10}, 40] (* Harvey P. Dale, Aug 23 2011 *)
Table[2^n -n-2, {n, 2, 40}] (* Eric W. Weisstein, Aug 09 2017 *)

Prog

(Maxima) A000247(n):=2^n-n-2$
makelist(A000247(n), n, 2, 30); /* Martin Ettl, Nov 08 2012 */
(PARI) a(n)=2^n-n-2 \\ Charles R Greathouse IV, Sep 28 2015
(Magma) [2^n -n-2: n in [2..40]]; // G. C. Greubel, Jul 26 2019
(Sage) [2^n -n-2 for n in (2..40)] # G. C. Greubel, Jul 26 2019
(GAP) List([2..40], n-> 2^n -n-2); # G. C. Greubel, Jul 26 2019

Crossrefs

Cf. A000478 (3 boxes), A058844 (4 boxes).

Sequence in context: A267574 A047667 A192963 * A097763 A034506 A067988

Adjacent sequences: A000244 A000245 A000246 * A000248 A000249 A000250

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Extensions

Additional comments from Michael Steyer, Dec 02 2000

More terms from Larry Reeves (larryr(AT)acm.org), Dec 04 2000

I recently changed the beginning of this sequence so the formulas etc. may need to be adjusted. - N. J. A. Sloane, Jan 24 2006

Formulas and comments adjusted for offset by Franklin T. Adams-Watters, Nov 10 2011

Status

approved