A239068 Triangle read by rows: row n lists the smallest positive ideal non-symmetric multigrade of degree n, or 2n+2 zeros if none.

1, 6, 9, 3, 3, 10, 1, 10, 12, 23, 3, 5, 16, 22, 1, 7, 17, 26, 30, 2, 5, 21, 22, 31, 1, 10, 18, 35, 37, 47, 2, 7, 25, 26, 43, 45, 1, 19, 20, 51, 57, 80, 82, 2, 12, 31, 40, 69, 71, 85, 1, 8, 24, 51, 54, 82, 83, 97, 2, 6, 27, 43, 64, 73, 89, 96

Offset: 1

Author

Jonathan Sondow, Mar 10 2014

Keywords

• hard
• more
• nonn
• tabf

Comments

The main entry for this topic is A239066.

A multigrade x1<=x2<=…<=xs; y1<=y2<=…<=ys is "symmetric" if x1+ys = x2+y(s-1) = … = xs+y1 when s is odd, or x1+xs = x2+x(s-1) = … = x(s/2)+x((s/2)+1) = y1+ys = y2+y(s-1) = … = y(s/2)+y((s/2)+1) when s is even. See A239067.

Any ideal multigrade x1,x2;y1,y2 of degree 1 is symmetric, since x1+x2 = y1+y2. Ideal non-symmetric multigrades are known only for degrees 2,3,4,5,6,7. The ones for degrees 5,6,7 are only conjecturally the smallest ones.

Examples

			1, 6, 9; 3, 3, 10
1, 10, 12, 23; 3, 5, 16, 22
1, 7, 17, 26, 30; 2, 5, 21, 22, 31
1, 10, 18, 35, 37, 47; 2, 7, 25, 26, 43, 45
1, 19, 20, 51, 57, 80, 82; 2, 12, 31, 40, 69, 71, 85
1, 8, 24, 51, 54, 82, 83, 97; 2, 6, 27, 43, 64, 73, 89, 96
1, 6, 9; 3, 3, 10 is an ideal non-symmetric multigrade of degree 2 as 1+10 != 6+3 and 1^1 + 6^1 + 9^1 = 16 = 3^1 + 3^1 + 10^1 and 1^2 + 6^2 + 9^2 = 118 = 3^2 + 3^2 + 10^2.
		

Crossrefs

Cf. A239066, A239067.

Formula

a(n^2 + n - 1) = 1 or 0.