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 A194628 Arises in enumerating Huffman codes, compact trees, and sums of unit fractions. 9
 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 Alois P. Heinz, Table of n, a(n) for n = 1..1000 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-1,4). - Alois P. Heinz, Nov 08 2017 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) 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[4n - 3, 1, 5]; Array[a, 40] (* Jean-François Alcover, Jul 21 2018, after Alois P. Heinz *) PROG (PARI) /* see A002572, set t=5 */ CROSSREFS Cf. A002572, A176485, A176503, A294775. Sequence in context: A332726 A239557 A001591 * A003240 A280543 A282566 Adjacent sequences: A194625 A194626 A194627 * A194629 A194630 A194631 KEYWORD nonn AUTHOR Jonathan Vos Post, Aug 30 2011 EXTENSIONS Terms beyond a(20)=113904 added by Joerg Arndt, Dec 18 2012 Invalid empirical g.f. removed by Alois P. Heinz, Nov 08 2017 STATUS approved

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Last modified April 1 22:04 EDT 2023. Contains 361705 sequences. (Running on oeis4.)