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%I #42 Mar 15 2017 16:24:09
%S 1,1,2,7,47,682,23132,1913821,397731998,212521309666,297464368728296
%N Median of terms of A254296 in the range (3^(n-1)+1)/2 to (3^n-1)/2.
%C As described in A254296, all the 'feasible' partitions of natural numbers (3^(n-1)+1)/2 to (3^n-1)/2 has n parts. A254439 lists the "median of the range ((3^(n-1)+1)/2)-th to ((3^n-1)/2)-th terms of A254296".
%C From conjectured formula, it appears that next terms are 1107102779611719118, 11090084422457163934046, 302002529294596303158583642. - _Benedict W. J. Irwin_, Nov 16 2016
%H Md Towhidul Islam & Md Shahidul Islam, <a href="http://arxiv.org/abs/1502.07730">Number of Partitions of an n-kilogram Stone into Minimum Number of Weights to Weigh All Integral Weights from 1 to n kg(s) on a Two-pan Balance</a>, arXiv:1502.07730 [math.CO], 2015.
%F a(n) = A254296(3^(n-1)).
%F Conjecture: for n>3, a(n+3) = Sum_{i_1=1..2} Sum_{i_2=1..3*i_1-1} ... Sum_{i_n..3*i_(n-1)-1} (3*i_n - 1). - _Benedict W. J. Irwin_, Nov 16 2016
%e As described in sequence A254296, "feasible" partitions of the integers 41 through 121 consist of 5 parts. The number 3^(5-1) = 81 has 47 "feasible" partitions, which is the median of the range from the 41st to the 121st term of A254296.
%t F[a_, x_, k_] := Sum[x, {a, 1, k}]
%t F[i1, 3*i1 - 1, 2]
%t F[i1, F[i2, 3*i2 - 1, 3*i1 - 1], 2]
%t F[i1, F[i2, F[i3, 3*i3 - 1, 3*i2 - 1], 3*i1 - 1], 2]
%t F[i1, F[i2, F[i3, F[i4, 3*i4 - 1, 3*i3 - 1], 3*i2 - 1], 3*i1 - 1], 2] (* Examples of how to get first few terms, use the C code to generate the n-th term of the conjectured formula, _Benedict W. J. Irwin_, Nov 16 2016 *)
%o (C)
%o /* C Code to make Mathematica Code for conjectured n-th term n>3 */
%o #include <stdio.h>
%o int main(int argc, char* argv[]){
%o int i, n=atoi(argv[1])-3;
%o printf("F[a_,x_,k_]:=Sum[x,{a,1,k}]\n");
%o for(i=1; i<=n; i++)printf("F[i%d,",i);
%o printf("3i%d-1,",n);
%o for(i=n-1; i>0; i--)printf("3i%d-1],",i);
%o printf("2]\n");
%o return 0;
%o }
%o /* _Benedict Irwin_, Nov 16 2016 */
%Y Cf. A254296, A254430, A254431, A254432, A254433, A254435, A254436, A254437, A254438, A254440.
%K nonn,more
%O 1,3
%A _Md. Towhidul Islam_, Mar 01 2015
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