%I #17 Oct 12 2021 21:54:07

%S 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,

%T 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,

%U 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

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

%C The main entry for this topic is A239066.

%C 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.

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

%H C. Starr, <a href="https://doi.org/10.1017/mag.2021.61">Notes on Listener Crossword 4595 by Elap</a>, The Mathematical Gazette (July 2021), Vol. 105, Issue 563, 291-298.

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

%e 1, 3; 2, 2

%e 1, 4, 4; 2, 2, 5

%e 1, 4, 5, 8; 2, 2, 7, 7

%e 1, 5, 9, 17, 18; 2, 3, 11, 15, 19

%e 1, 4, 6, 12, 14, 17; 2, 2, 9, 9, 16, 16

%e 1, 19, 28, 59, 65, 90, 102; 2, 14, 39, 45, 76, 85, 103

%e 1, 5, 10, 24, 28, 42, 47, 51; 2, 3, 12, 21, 31, 40, 49, 50

%e 1, 25, 31, 84, 87, 134, 158, 182, 198; 2, 18, 42, 66, 113, 116, 169, 175, 199

%e 1, 13, 126, 214, 215, 413, 414, 502, 615, 627; 6, 7, 134, 183, 243, 385, 445, 494, 621, 622

%e 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.

%e 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.

%Y Cf. A239066, A239068.

%K hard,nonn,tabf

%O 1,2

%A _Jonathan Sondow_, Mar 10 2014