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A176503 Leading column of triangle in A176463. 10
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

Table of n, a(n) for n=1..36.

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.

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)

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.

Sequence in context: A239555 A275544 A000078 * A262333 A217777 A034338

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

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Last modified June 23 02:51 EDT 2017. Contains 288633 sequences.