OFFSET
1,2
COMMENTS
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
REFERENCES
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).
LINKS
B. K. Agarwala and F. C. Auluck, Statistical mechanics and partitions into non-integral powers of integers, Proc. Camb. Phil. Soc., 47 (1951), 207-216.
B. K. Agarwala and F. C. Auluck, Statistical mechanics and partitions into non-integral powers of integers, Proc. Camb. Phil. Soc., 47 (1951), 207-216. [Annotated scanned copy]
EXAMPLE
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) }.
MAPLE
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
MATHEMATICA
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms from R. J. Mathar, Oct 19 2007
One more term from Herman Jamke (hermanjamke(AT)fastmail.fm), May 03 2008
a(20)-a(35) from Jon E. Schoenfield, Jan 17 2009
STATUS
approved