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 A176485 First column of triangle in A176452. 11
 1, 1, 1, 2, 4, 7, 13, 25, 48, 92, 176, 338, 649, 1246, 2392, 4594, 8823, 16945, 32545, 62509, 120060, 230598, 442910, 850701, 1633948, 3138339, 6027842, 11577747, 22237515, 42711863, 82037200, 157569867, 302646401, 581296715, 1116503866, 2144482948, 4118935248, 7911290530 (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) <= 3*p(k+1), see example.  [Joerg Arndt, Dec 18 2012] Row 2 of Table 1 of Elsholtz, row 1 being A002572. - Jonathan Vos Post, Aug 30 2011 LINKS Alois P. Heinz, Table of n, a(n) for n = 1..2000 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:1108.5964 [math.CO], Aug 30, 2011. Also IEEE Trans. Information Theory, Vol. 59, No. 2, 2013 pp. 1065-1075. FORMULA a(n) = A294775(n-1,2). - Alois P. Heinz, Nov 08 2017 EXAMPLE From Joerg Arndt, Dec 18 2012: (Start) There are a(7+1)=25 compositions 7=p(1)+p(2)+...+p(m) with p(1)=1 and p(k) <= 3*p(k+1): [ 1]  [ 1 1 1 1 1 1 1 ] [ 2]  [ 1 1 1 1 1 2 ] [ 3]  [ 1 1 1 1 2 1 ] [ 4]  [ 1 1 1 1 3 ] [ 5]  [ 1 1 1 2 1 1 ] [ 6]  [ 1 1 1 2 2 ] [ 7]  [ 1 1 1 3 1 ] [ 8]  [ 1 1 2 1 1 1 ] [ 9]  [ 1 1 2 1 2 ] [10]  [ 1 1 2 2 1 ] [11]  [ 1 1 2 3 ] [12]  [ 1 1 3 1 1 ] [13]  [ 1 1 3 2 ] [14]  [ 1 2 1 1 1 1 ] [15]  [ 1 2 1 1 2 ] [16]  [ 1 2 1 2 1 ] [17]  [ 1 2 1 3 ] [18]  [ 1 2 2 1 1 ] [19]  [ 1 2 2 2 ] [20]  [ 1 2 3 1 ] [21]  [ 1 2 4 ] [22]  [ 1 3 1 1 1 ] [23]  [ 1 3 1 2 ] [24]  [ 1 3 2 1 ] [25]  [ 1 3 3 ] (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[2n-1, 1, 3]; Array[a, 40] (* Jean-François 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=3;  /* 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. A176452, A002572, A176503, A294775. Sequence in context: A018083 A108361 A082423 * A119266 A102026 A103204 Adjacent sequences:  A176482 A176483 A176484 * A176486 A176487 A176488 KEYWORD nonn AUTHOR N. J. A. Sloane, Dec 07 2010 EXTENSIONS Extended by Jonathan Vos Post, Aug 30 2011 Added terms beyond a(20)=62509, Joerg Arndt, Dec 18 2012. STATUS approved

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Last modified May 9 12:48 EDT 2021. Contains 343740 sequences. (Running on oeis4.)