login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A176503 Leading column of triangle in A176463. 11
1, 1, 1, 2, 4, 8, 15, 29, 57, 112, 220, 432, 848, 1666, 3273, 6430, 12632, 24816, 48754, 95783, 188177, 369696, 726312, 1426930, 2803381, 5507590, 10820345, 21257915, 41763825, 82050242, 161197933, 316693445, 622183778, 1222357651, 2401474098, 4717995460 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

a(n+1) is the number of compositions n=p(1)+p(2)+...+p(m) with p(1)=1 and p(k) <= 4*p(k+1), see example.  [Joerg Arndt, Dec 18 2012]

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Christian Elsholtz, Clemens Heuberger, Daniel Krenn, Algorithmic counting of nonequivalent compact Huffman codes, arXiv:1901.11343 [math.CO], 2019.

Christian Elsholtz, Clemens Heuberger, Helmut Prodinger, The number of Huffman codes, compact trees, and sums of unit fractions, arXiv:1108.5964v1 [math.CO], Aug 30, 2011. Also IEEE Trans. Information Theory, Vol. 59, No. 2, 2013 pp. 1065-1075.

FORMULA

a(n) = A294775(n-1,3). - Alois P. Heinz, Nov 08 2017

EXAMPLE

From Joerg Arndt, Dec 18 2012: (Start)

There are a(6+1)=15 compositions 6=p(1)+p(2)+...+p(m) with p(1)=1 and p(k) <= 4*p(k+1):

[ 1]  [ 1 1 1 1 1 1 ]

[ 2]  [ 1 1 1 1 2 ]

[ 3]  [ 1 1 1 2 1 ]

[ 4]  [ 1 1 1 3 ]

[ 5]  [ 1 1 2 1 1 ]

[ 6]  [ 1 1 2 2 ]

[ 7]  [ 1 1 3 1 ]

[ 8]  [ 1 1 4 ]

[ 9]  [ 1 2 1 1 1 ]

[10]  [ 1 2 1 2 ]

[11]  [ 1 2 2 1 ]

[12]  [ 1 2 3 ]

[13]  [ 1 3 1 1 ]

[14]  [ 1 3 2 ]

[15]  [ 1 4 1 ]

(End)

MATHEMATICA

b[n_, r_, k_] := b[n, r, k] = If[n < r, 0, If[r == 0, If[n == 0, 1, 0], Sum[b[n-j, k*(r-j), k], {j, 0, Min[n, r]}]]];

a[n_] := b[3n-2, 1, 4];

Array[a, 40] (* Jean-Fran├žois Alcover, Jul 21 2018, after Alois P. Heinz *)

PROG

(PARI)

/* g.f. as given in the Elsholtz/Heuberger/Prodinger reference */

N=66;  q='q+O('q^N);

t=4;  /* t-ary: t=2 for A002572, t=3 for A176485, t=4 for A176503  */

L=2 + 2*ceil( log(N) / log(t) );

f(k) = (1-t^k)/(1-t);

la(j) = prod(i=1, j, q^f(i) / ( 1 - q^f(i) ) );

nm=sum(j=0, L, (-1)^j * q^f(j) * la(j) );

dn=sum(j=0, L, (-1)^j * la(j) );

gf = nm / dn;

Vec( gf )

/* Joerg Arndt, Dec 27 2012 */

CROSSREFS

Cf. A176463, A194628 - A194633, A294775.

Sequence in context: A239555 A275544 A000078 * A262333 A293335 A301480

Adjacent sequences:  A176500 A176501 A176502 * A176504 A176505 A176506

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Dec 07 2010

EXTENSIONS

Added terms beyond a(13)=848, Joerg Arndt, Dec 18 2012

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified October 18 03:22 EDT 2019. Contains 328135 sequences. (Running on oeis4.)