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A000234
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Partitions into non-integral powers (see Comments for precise definition).
(Formerly M2730 N1095)
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3
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1, 3, 8, 18, 37, 72, 136, 251, 445, 770, 1312, 2202, 3632, 5908, 9501, 15111, 23781, 37083, 57293, 87813, 133530, 201574, 302265, 450317, 666743, 981488, 1437003, 2092976, 3033253, 4375104, 6282026, 8981046, 12786327, 18131492, 25612628
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OFFSET
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1,2
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COMMENTS
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This sequence gives the number of solutions to the inequality Sum_{i=1,2,...} xi^(2/3) <= n with the constraint that 1 <= x1 <= x2 <= x3 <= ... is a list of at least 1 and no more than n integers. - R. J. Mathar, Oct 19 2007
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REFERENCES
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N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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EXAMPLE
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a(3)=8 counts 5 partitions with 1 term, explicitly { 1^(2/3), 2^(2/3), 3^(2/3), 4^(2/3), 5^(2/3) }, 2 partitions into sums of 2 terms { 1^(2/3) + 1^(2/3), 1^(2/3) + 2^(2/3) } and one partition into a sum of three terms { 1^(2/3) + 1^(2/3) + 1^(2/3) }.
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MAPLE
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fs:=n->floor(simplify(n)): a:=proc(i, m, k) options remember: local s, l, j, m2: if(k=1) then RETURN(1) else s:=0: l:=fs(m^(3/2)): for j from 1 to min(l, i) do m2:=m-j^(2/3): if(fs(m2)>=1) then s:=s+a(j, m2, k-1) fi: s:=s+1 od: RETURN(s) fi: end: seq(a(fs(n^(3/2)), n, n), n=1..19); # Herman Jamke (hermanjamke(AT)fastmail.fm), May 03 2008
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MATHEMATICA
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fs[n_] := Floor[Simplify[n]]; a[i_, m_, k_] := a[i, m, k] = Module[{s, l, j, m2}, If[k == 1, Return[1], s = 0; l = fs[m^(3/2)]; For[j = 1, j <= Min[l, i], j++, m2 = m - j^(2/3); If[fs[m2] >= 1, s = s + a[j, m2, k-1] ]; s = s+1]; Return[s]]]; A000234 = Table[an = a[fs[n^(3/2)], n, n]; Print["a(", n, ") = ", an]; an, {n, 1, 19}] (* Jean-François Alcover, Feb 06 2016, after Herman Jamke *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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One more term from Herman Jamke (hermanjamke(AT)fastmail.fm), May 03 2008
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STATUS
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approved
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