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%I M2730 N1095 #31 Oct 02 2017 02:12:54
%S 1,3,8,18,37,72,136,251,445,770,1312,2202,3632,5908,9501,15111,23781,
%T 37083,57293,87813,133530,201574,302265,450317,666743,981488,1437003,
%U 2092976,3033253,4375104,6282026,8981046,12786327,18131492,25612628
%N Partitions into non-integral powers (see Comments for precise definition).
%C 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
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H B. K. Agarwala and F. C. Auluck, <a href="http://dx.doi.org/10.1017/S0305004100026505">Statistical mechanics and partitions into non-integral powers of integers</a>, Proc. Camb. Phil. Soc., 47 (1951), 207-216.
%H B. K. Agarwala and F. C. Auluck, <a href="/A000093/a000093.pdf">Statistical mechanics and partitions into non-integral powers of integers</a>, Proc. Camb. Phil. Soc., 47 (1951), 207-216. [Annotated scanned copy]
%e 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) }.
%p 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
%t 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 *)
%K nonn
%O 1,2
%A _N. J. A. Sloane_
%E More terms from _R. J. Mathar_, Oct 19 2007
%E One more term from Herman Jamke (hermanjamke(AT)fastmail.fm), May 03 2008
%E a(20)-a(35) from _Jon E. Schoenfield_, Jan 17 2009
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