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A239067
Triangle read by rows: row n lists the smallest positive ideal symmetric multigrade of degree n, or 2n+2 zeros if none.
5
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
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
Sequence in context: A262672 A252731 A239066 * A131015 A342769 A130195
KEYWORD
hard,nonn,tabf
AUTHOR
Jonathan Sondow, Mar 10 2014
STATUS
approved