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A000234 Partitions into non-integral powers (see Comments for precise definition).
(Formerly M2730 N1095)
2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

This 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

Table of n, a(n) for n=1..35.

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-Fran├žois Alcover, Feb 06 2016, after _Herman Jamke_ *)

CROSSREFS

Sequence in context: A169763 A227161 A241080 * A136376 A099845 A036635

Adjacent sequences:  A000231 A000232 A000233 * A000235 A000236 A000237

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

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

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Last modified March 26 00:51 EDT 2017. Contains 284111 sequences.