A239067 Triangle read by rows: row n lists the smallest positive ideal symmetric multigrade of degree n, or 2n+2 zeros if none.

1, 3, 2, 2, 1, 4, 4, 2, 2, 5, 1, 4, 5, 8, 2, 2, 7, 7, 1, 5, 9, 17, 18, 2, 3, 11, 15, 19, 1, 4, 6, 12, 14, 17, 2, 2, 9, 9, 16, 16, 1, 19, 28, 59, 65, 90, 102, 2, 14, 39, 45, 76, 85, 103, 1, 5, 10, 24, 28, 42, 47, 51, 2, 3, 12, 21, 31, 40, 49, 50, 1, 25, 31, 84, 87, 134, 158, 182, 198, 2, 18, 42, 66, 113, 116, 169, 175, 199, 1, 13, 126, 214, 215, 413, 414, 502, 615, 627, 6, 7, 134, 183, 243, 385, 445, 494, 621, 622

Offset

1,2

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. For non-symmetric ones, see A239068.

The ideal symmetric multigrades of degrees 5,6,7,8,9,10 are only conjecturally the smallest ones.

Links

Table of n, a(n) for n=1..108.

C. Starr, Notes on Listener Crossword 4595 by Elap, The Mathematical Gazette (July 2021), Vol. 105, Issue 563, 291-298.

Formula

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

Example

1, 3; 2, 2
1, 4, 4; 2, 2, 5
1, 4, 5, 8; 2, 2, 7, 7
1, 5, 9, 17, 18; 2, 3, 11, 15, 19
1, 4, 6, 12, 14, 17; 2, 2, 9, 9, 16, 16
1, 19, 28, 59, 65, 90, 102; 2, 14, 39, 45, 76, 85, 103
1, 5, 10, 24, 28, 42, 47, 51; 2, 3, 12, 21, 31, 40, 49, 50
1, 25, 31, 84, 87, 134, 158, 182, 198; 2, 18, 42, 66, 113, 116, 169, 175, 199
1, 13, 126, 214, 215, 413, 414, 502, 615, 627; 6, 7, 134, 183, 243, 385, 445, 494, 621, 622
1, 4, 4; 2, 2, 5 is an ideal symmetric multigrade of degree 2 as 1+5 = 4+2 = 4+2 and 1^1 + 4^1 + 4^1 = 9 = 2^1 + 2^1 + 5^1 and 1^2 + 4^2 + 4^2 = 33 = 2^2 + 2^2 + 5^2.
1, 4, 5, 8; 2, 2, 7, 7 is an ideal symmetric multigrade of degree 3 as 1+8 = 4+5 = 2+7 = 2+7 and 1^1 + 4^1 + 5^1 + 8^1 = 18 = 2^1 + 2^1 + 7^1 + 7^1 and 1^2 + 4^2 + 5^2 + 8^2 = 106 = 2^2 + 2^2 + 7^2 + 7^2 and 1^3 + 4^3 + 5^3 + 8^3 = 702 = 2^3 + 2^3 + 7^3 + 7^3.

Crossrefs

Cf. A239066, A239068.

Sequence in context: A262672 A252731 A239066 * A131015 A342769 A130195

Adjacent sequences: A239064 A239065 A239066 * A239068 A239069 A239070

Keyword

hard,nonn,tabf

Author

Jonathan Sondow, Mar 10 2014

Status

approved