A104185
Number of partitions of the set 1, 2, 3, ..., 6n+3 into 2n+1 sets of 3 elements each, such that each 3-element set has the same sum (there are no such partitions unless there are 6n+3 elements).
Original entry on oeis.org
1, 2, 11, 84, 1296, 24293, 703722, 24212879, 1157746949, 63552536107
Offset: 0
a(1) = 2 because with 9 elements they can be partitioned (9 5 1) (8 4 3) (7 6 2) or (9 4 2) (8 6 1) (7 5 3)
- Dossey, Giordano, McCrone and Weir, Mathematics methods and modeling for today's mathematics classroom, p. 134
More terms from Guenter Stertenbrink
The terms 1157746949, 63552536107 were found by
Don Knuth, Sep 04 2009
A168238
Number of different 0-moment rowing configurations for 4n rowers.
Original entry on oeis.org
1, 4, 29, 263, 2724, 30554, 361677, 4454273, 56546511, 735298671, 9749613914, 131377492010, 1794546880363, 24798396567242, 346130144084641, 4873560434459530, 69149450121948083, 987844051312409668, 14198028410251734447, 205181815270346718199
Offset: 1
For n = 2 there are 4 solutions to the 8-man rowing problem.
For n=1 the unique solution is 1+4 = 2+3. For n=2 there are 4 different solutions: 1+2+7+8 = 3+4+5+6, 1+3+6+8 = 2+4+5+7, 1+4+5+8 = 2+3+6+7, 1+4+6+7 = 2+3+5+8. - _Michel Marcus_, May 25 2013
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b[L_List] := b[L] = Module[{nL = Length[L], k = L[[-1]], m = L[[-2]]}, Which[L[[1]] == 0, If[nL == 3, 1, b[L[[2 ;; nL]]]], L[[1]] < 1, 0, True, Sum[If[L[[j]] < m, 0, b[Join[Sort[Table[L[[i]] - If[i == j, m + 1/97, 0], {i, 1, nL - 2}]], {m - 1, k}]]], {j, 1, nL - 2}]]];
A[n_, k_] := If[n==0 || k==0, 1, b[Join[Array[(k (n k + 1)/2 + k/97)&, n], {k n, k}]]/n!];
a[n_] := A[2, 2n];
Array[a, 20] (* Jean-François Alcover, Aug 19 2018, after Alois P. Heinz *)
A203435
Number of partitions of {1,2,...,4n} into n 4-element subsets having the same sum.
Original entry on oeis.org
1, 1, 4, 32, 392, 6883, 171088, 5661874, 242038179, 13147317481
Offset: 0
a(1) = 1: {1,2,3,4}.
a(2) = 4: {1,2,7,8}, {3,4,5,6}; {1,3,6,8}, {2,4,5,7}; {1,4,5,8}, {2,3,6,7}; {1,4,6,7}, {2,3,5,8}.
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b:= proc() option remember; local i, j, t, m; m:= args[nargs]; if args[1]=0 then `if`(nargs=2, 1, b(args[t] $t=2..nargs)) elif args[1]<1 then 0 else add(`if`(args[j] `if`(n=0, 1, b(((8*n+2)+4/97) $n, 4*n)/n!): seq(a(n), n=0..6);
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b[l_] := b[l] = Module[{nl = Length[l], k = l[[-1]], m = l[[-2]]}, Which[l[[1]] == 0, If[nl == 3, 1, b[l[[2 ;; nl]]]], l[[1]] < 1, 0, True, Sum[If[l[[j]] < m, 0, b[Join[Sort[Table[l[[i]] - If[i == j, m + 1/97, 0], {i, 1, nl - 2}]], {m - 1, k}]]], {j, 1, nl - 2}]]];
a[n_] := If[n == 0, 1, b[Join[Array[8*n + 2 + 4/97& , n], {4*n, 4}]]/n!];
Table[a[n], {n, 0, 6}] (* Jean-François Alcover, Jun 03 2018, adapted from Maple *)
A203017
Number of partitions of {1,2,...,3n} into 3 n-element subsets having the same sum.
Original entry on oeis.org
1, 0, 1, 2, 32, 305, 4331, 63261, 1025113, 17495345, 313692810, 5838204047, 112185853894, 2213711510395, 44691175805738, 920173212324164, 19274796589413439, 409908483736507979, 8835309887111026335, 192739853119591626715, 4250191938786946069812, 94641409538083474973850
Offset: 0
a(0) = 1: {}, {}, {}.
a(1) = 0: there is no partition of {1,2,3} into 3 1-element subsets having the same sum.
a(2) = 1: {1,6}, {2,5}, {3,4}.
a(3) = 2: {1,5,9}, {2,6,7}, {3,4,8}; {1,6,8}, {2,4,9}, {3,5,7}.
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b:= proc() option remember; local i, j, t, k, m; m:= args[nargs-1]; k:= args[nargs-0]; if args[1]=0 then `if`(nargs=3, 1, b(args[t]$t=2..nargs)) elif args[1]<1 then 0 else add(`if`(args[j] b((n*(3*n+1)/2 +n/97)$3, 3*n, n)/`if`(n>0, 6, 1):
seq(a(n), n=0..10);
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b[args_] := b[args] = Module[{nargs = Length[args], k = args[[-1]], m = args[[-2]]}, Which[args[[1]] == 0, If[nargs == 3, 1, b[args[[2 ;; nargs]] ]], args[[1]]<1, 0, True, Sum[If[args[[j]] < m, 0, b[Join[Sort[Table[ args[[i]] - If[i == j, m + 1/97, 0], {i, 1, nargs - 2}]], {m - 1, k}]]], {j, 1, nargs-2}]]];
A[n_, k_] := If[n == 0 || k == 0, 1, b[Join[Array[(k*(n*k + 1)/2 + k/97)&, n], {k*n, k}]]/n!];
a[k_] := A[3, k];
a /@ Range[0, 10] (* Jean-François Alcover, Dec 05 2020, after Alois P. Heinz *)
A321230
Number of set partitions of [n^2] into n n-element subsets having the same sum.
Original entry on oeis.org
1, 1, 1, 2, 392, 3245664, 6534071578530
Offset: 0
a(0) = 1: empty.
a(1) = 1: 1.
a(2) = 1: 14|23.
a(3) = 2: 168|249|357, 159|267|348.
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