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A079500 Number of compositions of the integer n in which the first part is >= the other parts. 22
1, 1, 2, 3, 5, 8, 14, 24, 43, 77, 140, 256, 472, 874, 1628, 3045, 5719, 10780, 20388, 38674, 73562, 140268, 268066, 513350, 984911, 1892875, 3643570, 7023562, 13557020, 26200182, 50691978, 98182666, 190353370, 369393466, 717457656, 1394632365, 2713061899 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Essentially the same as A007059: a(n) = A007059(n+1).

In lunar arithmetic in base 2, this is the number of lunar divisors of the number 111...1 (with n 1's). E.g. 1111 has a(4) = 5 divisors (see A048888). - N. J. A. Sloane, Feb 23 2011.

First differences of A186537. - N. J. A. Sloane, Feb 23 2011

Number of balanced ordered rooted trees with n non-root nodes (see A048816 for unordered balanced trees); see example. The compositions are obtained from the level sequences by identifying a length-k run of (non-root) levels [t, t+1, t+2, ..., t+k-1] with a part k. - Joerg Arndt, Jul 20 2014

REFERENCES

Arnold Knopfmacher and Neville Robbins, Compositions with parts constrained by the leading summand, Ars Combin. 76 (2005), 287-295.

LINKS

T. D. Noe and Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 400 terms from T. D. Noe)

D. Applegate, M. LeBrun and N. J. A. Sloane, Dismal Arithmetic [Note: we have now changed the name from "dismal arithmetic" to "lunar arithmetic" - the old name was too depressing]

Srecko Brlek, Andrea Frosini, Simone Rinaldi, Laurent Vuillon, Tilings by translation: enumeration by a rational language approach, The Electronic Journal of Combinatorics, vol.13, (2006).

A. Frosini and S. Rinaldi, On the Sequence A079500 and Its Combinatorial Interpretations, Journal of Integer Sequences, Vol. 9 (2006), Article 06.3.1.

R. Kemp, Balanced ordered trees, Random Structures Algorithms, 5 (1994), pp. 99-121.

Index entries for sequences related to dismal (or lunar) arithmetic

FORMULA

G.f.: (1-z) * sum(k>=0, z^k/(1-2*z+z^(k+1)) ).

a(n) = A048888(n) - 1.

This is a subsequence of A067399: a(n) = A067399(2^n-1).

G.f.: -(1+x^2+ 1/(x-1) )/x*( 1 + x*(x-1)^3*(1-x+x^3)/( Q(0)- x*(x-1)^3*(1-x+x^3)) ), where Q(k) = (x+1)*(2*x-1)*(1-x)^2 + x^(k+2)*(x+x^2+x^3-2*x^4-1 - x^(k+3) + x^(k+5)) - x*(-1+2*x-x^(k+3))*(1-2*x+x^2+x^(k+4)-x^(k+5))*(-1+4*x-5*x^2+2*x^3 - x^(k+2)- x^(k+5) + 2*x^(k+3) - x^(2*k+5) + x^(2*k+6))/Q(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Dec 14 2013

EXAMPLE

From Joerg Arndt, Dec 29 2012: (Start)

There are a(7)=24 compositions p(1)+p(2)+...+p(m)=7 such that p(k)<=p(1):

[ 1]  [ 1 1 1 1 1 1 1 ]

[ 2]  [ 2 1 1 1 1 1 ]

[ 3]  [ 2 1 1 1 2 ]

[ 4]  [ 2 1 1 2 1 ]

[ 5]  [ 2 1 2 1 1 ]

[ 6]  [ 2 1 2 2 ]

[ 7]  [ 2 2 1 1 1 ]

[ 8]  [ 2 2 1 2 ]

[ 9]  [ 2 2 2 1 ]

[10]  [ 3 1 1 1 1 ]

[11]  [ 3 1 1 2 ]

[12]  [ 3 1 2 1 ]

[13]  [ 3 1 3 ]

[14]  [ 3 2 1 1 ]

[15]  [ 3 2 2 ]

[16]  [ 3 3 1 ]

[17]  [ 4 1 1 1 ]

[18]  [ 4 1 2 ]

[19]  [ 4 2 1 ]

[20]  [ 4 3 ]

[21]  [ 5 1 1 ]

[22]  [ 5 2 ]

[23]  [ 6 1 ]

[24]  [ 7 ]

(End)

From Joerg Arndt, Jul 20 2014: (Start)

The a(7) = 24 balanced ordered rooted trees with 7 non-root nodes are, as level sequences (of the pre-order walk):

01:  [ 0 1 1 1 1 1 1 1 ]

02:  [ 0 1 2 1 2 1 2 2 ]

03:  [ 0 1 2 1 2 2 1 2 ]

04:  [ 0 1 2 1 2 2 2 2 ]

05:  [ 0 1 2 2 1 2 1 2 ]

06:  [ 0 1 2 2 1 2 2 2 ]

07:  [ 0 1 2 2 2 1 2 2 ]

08:  [ 0 1 2 2 2 2 1 2 ]

09:  [ 0 1 2 2 2 2 2 2 ]

10:  [ 0 1 2 3 1 2 3 3 ]

11:  [ 0 1 2 3 2 3 2 3 ]

12:  [ 0 1 2 3 2 3 3 3 ]

13:  [ 0 1 2 3 3 1 2 3 ]

14:  [ 0 1 2 3 3 2 3 3 ]

15:  [ 0 1 2 3 3 3 2 3 ]

16:  [ 0 1 2 3 3 3 3 3 ]

17:  [ 0 1 2 3 4 2 3 4 ]

18:  [ 0 1 2 3 4 3 4 4 ]

19:  [ 0 1 2 3 4 4 3 4 ]

20:  [ 0 1 2 3 4 4 4 4 ]

21:  [ 0 1 2 3 4 5 4 5 ]

22:  [ 0 1 2 3 4 5 5 5 ]

23:  [ 0 1 2 3 4 5 6 6 ]

24:  [ 0 1 2 3 4 5 6 7 ]

(End)

MAPLE

M:=101:

t1:=add( (1-x)*x^k/(1-2*x+x^k), k=1..M):

series(t1, x, M-1);

seriestolist(%);

# second Maple program:

b:= proc(n, m) option remember; `if`(n=0, 1,

      `if`(m=0, add(b(n-j, j), j=1..n),

      add(b(n-j, min(n-j, m)), j=1..min(n, m))))

    end:

a:= n-> b(n, 0):

seq(a(n), n=0..40);  # Alois P. Heinz, May 01 2014

MATHEMATICA

nn=36; CoefficientList[Series[Sum[x^i/(1-(x-x^(i+1))/(1-x)), {i, 0, nn}], {x, 0, nn}], x]  (* Geoffrey Critzer, Mar 12 2013 *)

b[n_, m_] := b[n, m] = If[n==0, 1, If[m==0, Sum[b[n-j, j], {j, 1, n}], Sum[ b[n-j, Min[n-j, m]], {j, 1, Min[n, m]}]]]; a[n_] := b[n, 0]; Table[a[n], {n, 0, 40}] (* Jean-Fran├žois Alcover, Nov 23 2015, after Alois P. Heinz *)

CROSSREFS

Cf. A188541.

Sequence in context: A108351 A038495 A007059 * A108296 A072100 A104882

Adjacent sequences:  A079497 A079498 A079499 * A079501 A079502 A079503

KEYWORD

nonn

AUTHOR

Arnold Knopfmacher, Jan 21 2003

EXTENSIONS

Offset corrected by N. J. A. Sloane, Feb 23 2011

More terms from N. J. A. Sloane, Feb 24 2011

Further edits (required in order to clarify the definition - is the first part >= the rest. or only > the rest? Answer: the former; for the latter, see A007059) by N. J. A. Sloane, May 08 2011

STATUS

approved

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Last modified June 28 11:48 EDT 2017. Contains 288822 sequences.