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A194628 Arises in enumerating Huffman codes, compact trees, and sums of unit fractions. 7
1, 1, 1, 2, 4, 8, 16, 31, 61, 121, 240, 476, 944, 1872, 3712, 7362, 14601, 28958, 57432, 113904, 225904, 448034, 888583, 1762321, 3495200, 6932008, 13748208, 27266738, 54077957, 107252486, 212713209, 421872826, 836697836, 1659417786, 3291113315, 6527245245, 12945446241, 25674625681 (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) <= 5*p(k+1), see example.  [Joerg Arndt, Dec 18 2012]

Row 4 of Table 1 of Elsholtz, row 1 being A002572, row 2 being A176485, and row 3 being A176503.

LINKS

Table of n, a(n) for n=1..38.

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.

FORMULA

Empirical g.f.: x*(x^13+x^8+x^7-x^6-x^2-x+1) / (3*x^7-x^6-2*x+1). - Colin Barker, May 09 2013

EXAMPLE

From Joerg Arndt, Dec 18 2012: (Start)

There are a(6+1)=16 compositions 6=p(1)+p(2)+...+p(m) with p(1)=1 and p(k) <= 5*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 ]

[16]  [ 1 5 ]

(End)

PROG

(PARI) /* see A002572, set t=5 */

CROSSREFS

Cf. A002572, A176485, A176503.

Sequence in context: A128761 A239557 A001591 * A003240 A280543 A282566

Adjacent sequences:  A194625 A194626 A194627 * A194629 A194630 A194631

KEYWORD

nonn

AUTHOR

Jonathan Vos Post, Aug 30 2011

EXTENSIONS

Added terms beyond a(20)=113904, Joerg Arndt, Dec 18 2012.

STATUS

approved

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Last modified March 27 22:02 EDT 2017. Contains 284182 sequences.