A382400
Number of subsets of Z_n such that every ordered pair of distinct elements has a different sum.
Original entry on oeis.org
1, 2, 4, 8, 15, 26, 48, 78, 133, 202, 316, 474, 755, 1054, 1604, 2196, 3305, 4370, 6208, 8228, 11631, 15086, 20912, 26842, 37581, 46626, 64052, 79984, 109635, 133314, 176156, 217094, 291409, 343872, 457828, 547576, 718375, 852074, 1112128, 1308230, 1714741
Offset: 0
The a(6) = 48 subsets are 42 subsets of size at most 3 and the following 6: {1,3,4,5}, {1,2,3,5}, {0,2,4,5}, {0,2,3,4}, {0,1,3,5}, {0,1,2,4}. Each of the size 4 subsets is perfect in the sense that every number from 0..5 can be written as the sum of two elements modulo 6 in exactly one way.
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a(n)={
my(recurse(k,r,b,w)=
if(k >= n, 1,
my(t=bitand((1<
A169948
Fourth entry in row n of triangle in A169945.
Original entry on oeis.org
1, 2, 6, 14, 29, 52, 96, 160, 277, 450, 712, 1086, 1657, 2448, 3636, 5280, 7635, 10840, 15392, 21372, 29655, 40580, 55282, 74620, 100651, 134232, 178922, 236488, 312019, 408550, 534288, 692978, 897931, 1156256, 1485650, 1897704, 2421635, 3071608, 3894042
Offset: 2
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b[n_, s_] := b[n, s] = Module[{sn, m}, m = Length[s]; sn = Append[s, n]; If[n < 1, 1, b[n - 1, s] + If[m*(m + 1)/2 == Length[Union[Flatten[Table[ sn[[i]] + sn[[j]], {i, 1, m}, {j, i + 1, m + 1}]]]], b[n - 1, sn], 0]]];
A196723[n_] := A196723[n] = b[n - 1, {n}] + If[n == 0, 0, A196723[n - 1]];
c[n_, s_] := c[n, s] = Module[{sn, m}, If[n < 1, 1, sn = Append[s, n]; m = Length[sn]; If[m*(m - 1)/2 == Length[Table[sn[[i]] - sn[[j]], {i, 1, m - 1}, {j, i + 1, m}] // Flatten // Union], c[n - 1, sn], 0] + c[n-1, s]]];
A143823[n_] := A143823[n] = c[n - 1, {n}] + If[n == 0, 0, A143823[n - 1]];
a[n_] := a[n] = A196723[n + 1] - A143823[n + 1];
Table[Print[n, " ", a[n]]; a[n], {n, 2, 40}] (* Jean-François Alcover, Aug 27 2019, after Alois P. Heinz in A196723 and A143823 *)
A169953
Third entry in row n of triangle in A169950.
Original entry on oeis.org
1, 1, 4, 8, 15, 23, 44, 64, 117, 173, 262, 374, 571, 791, 1188, 1644, 2355, 3205, 4552, 5980, 8283, 10925, 14702, 19338, 26031, 33581, 44690, 57566, 75531, 96531, 125738, 158690, 204953, 258325, 329394, 412054, 523931, 649973, 822434, 1018332, 1274909
Offset: 2
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b[n_, s_] := b[n, s] = Module[{sn, m}, m = Length[s]; sn = Append[s, n]; If[n < 1, 1, b[n - 1, s] + If[m*(m + 1)/2 == Length[Union[Flatten[Table[ sn[[i]] + sn[[j]], {i, 1, m}, {j, i + 1, m + 1}]]]], b[n - 1, sn], 0]]];
A196723[n_] := A196723[n] = b[n - 1, {n}] + If[n == 0, 0, A196723[n - 1]]; c[n_, s_] := c[n, s] = Module[{sn, m}, If[n < 1, 1, sn = Append[s, n]; m = Length[sn]; If[m*(m - 1)/2 == Length[Table[sn[[i]] - sn[[j]], {i, 1, m - 1}, {j, i + 1, m}] // Flatten // Union], c[n-1, sn], 0] + c[n-1, s]]];
A143823[n_] := A143823[n] = c[n - 1, {n}] + If[n == 0, 0, A143823[n - 1]];
a[n_] := A196723[n+1] - A196723[n] - A143823[n+1] + A143823[n];
Table[Print[n, " ", a[n]]; a[n], {n, 2, 42}] (* Jean-François Alcover, Sep 07 2019, after Alois P. Heinz in A196723 and A143823 *)
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