A274103 Erroneous version of A003323.
3, 6, 17, 66, 327
Offset: 1
This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
Golomb and Baumert find a(4) = 44 and give this example:
A = {1, 3, 5, 15, 17, 19, 26, 28, 40, 42, 44}
B = {2, 7, 8, 18, 21, 24, 27, 37, 38, 43}
C = {4, 6, 13, 20, 22, 23, 25, 30, 32, 39, 41}
D = {9, 10, 11, 12, 14, 16, 29, 31, 33, 34, 35, 36}
Note that the union of these sets is {1, ..., 44} and none of the sets contains three numbers (perhaps not all distinct) such that one is the sum of the other two. - _Charles R Greathouse IV_, Jun 11 2013
From _Marijn Heule_, Nov 26 2017: (Start)
Exoo computed the first certificate showing that a(5) >= 160:
A = {1, 6, 10, 18, 21, 23, 26, 30, 34, 38, 43, 45, 50, 54, 65, 74, 87, 96, 107, 111, 116, 118, 123, 127, 131, 135, 138, 140, 143, 151, 155, 160}
B = {2, 3, 8, 14, 19, 20, 24, 25, 36, 46, 47, 51, 62, 73, 88, 99, 110, 114, 115, 125, 136, 137, 141, 142, 147, 153, 158, 159}
C = {4, 5, 15, 16, 22, 28, 29, 39, 40, 41, 42, 48, 49, 59, 102, 112, 113, 119, 120, 121, 122, 132, 133, 139, 145, 146, 156, 157}
D = {7, 9, 11, 12, 13, 17, 27, 31, 32, 33, 35, 37, 53, 56, 57, 61, 79, 82, 100, 104, 105, 108, 124, 126, 128, 129, 130, 134, 144, 148, 149, 150, 152, 154}
E = {44, 52, 55, 58, 60, 63, 64, 66, 67, 68, 69, 70, 71, 72, 75, 76, 77, 78, 80, 81, 83, 84, 85, 86, 89, 90, 91, 92, 93, 94, 95, 97, 98, 101, 103, 106, 109, 117} (End)
a:= proc(n) a(n):= `if`(n=0, 2, n*a(n-1)-n+2) end: seq(a(n), n=0..30); # Alois P. Heinz, Feb 17 2014
f[n_] := n*(f[n - 1] - 1) + 2;f[0]=2; ff[n_]:=(1/(1+n))(1+E*Gamma[1+n, 1]-E*(n^2)*Gamma[1+n, 1]+E*n*Gamma[2+n, 1]) (Spindler)
Table[FunctionExpand[Gamma[n, 1] E] + 1, {n, 2, 29}] (* Vincenzo Librandi, Feb 17 2014 *)
For n = 0 and n = 1, the empty graph K_0 and the singleton graph K_1 don't have any edge, so zero colors are needed. For n = 2 we have one edge, so one color is needed. For n = 3 we have a triangle, so we need a second color for the third edge. For n = 4 (square + diagonals) and n = 5 (pentagon + diagonals forming a pentagram) two colors are still enough: one can use one color for the "circumference", i.e., edges (i,i+1), and the other color for the diagonals. For n >= 6, a third color is needed.
A343607(n)=if(n>1, vecmax(color(n))+1, 0) \\ using the helper function: {M343607=List([[]]); color(n, i=matrix(n,n,r,c,r+c--*c--/2), C, k)= C|| C = if(#M343607 >= n, M343607[n], n>2, color(n-1,i)); k|| k=if(n>3, vecmax(C)+1, n-1); C=Vec(C, n*(n-1)/2); my(bad(C)= for(a=1,n-2, my(c=C[i[a,n]]); for(b=a+1, n-1, C[i[a,b]] !=c || C[i[b,n]] !=c || return( i[b,n] ))), C0=C, j); while(j=bad(C), until(j-- < i[1,n], if(C[j]++ < k, while(j<#C, C[j++]=0); next(2))); while(C[j]++ >= k, C[j]=0; j--); C=Vec(color(n-1,i,C[1..-n]),#C); if(C[1..n] != C0[1..n], k++; C=C0)); #M343607= 13. Changing 1..n to 1..2-2*n is much faster but yields suboptimal solution for n >= 12 (using 4 instead of 3 required colors).
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