A202670 Symmetric matrix based on A000290 (the squares), by antidiagonals.

1, 4, 4, 9, 17, 9, 16, 40, 40, 16, 25, 73, 98, 73, 25, 36, 116, 184, 184, 116, 36, 49, 169, 298, 354, 298, 169, 49, 64, 232, 440, 584, 584, 440, 232, 64, 81, 305, 610, 874, 979, 874, 610, 305, 81, 100, 388, 808, 1224, 1484, 1484, 1224, 808, 388, 100, 121

Offset: 1

Author

Clark Kimberling, Dec 22 2011

Keywords

• nonn
• tabl

Comments

Let s=(1,4,9,16,...) and let T be the infinite square matrix whose n-th row is formed by putting n-1 zeros before the terms of s. Let T' be the transpose of T. Then A202670 represents the matrix product M=T'*T. M is the self-fusion matrix of s, as defined at A193722. See A202671 for characteristic polynomials of principal submatrices of M.

...

row 1 (1,4,9,16,...) A000290

row 2 (4,17,40,73,...) A145995

diagonal (1,17,98,354,...) A000538

antidiagonal sums (1,8,35,112,...) A040977

...

The n-th "square border sum" m(n,1)+m(n,2)+...+m(n,n)+m(n-1,n)+m(n-2,n)+...+m(1,n) is a squared square pyramidal number: [n*(n+1)*(2*n+1)/6]^2; see A000330.

Examples

			Northwest corner:
1.....4......9....16....25
4....17.....40....73...116
9....40.....98...184...298
16...73....184...354...584
25...116...298...584...979
		

Crossrefs

Cf. A000290, A202671, A193722.

Programs

Mathematica

U = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[ Table[k^2, {k, 1, 12}]];
L = Transpose[U]; M = L.U; TableForm[M]
m[i_, j_] := M[[i]][[j]];
Flatten[Table[m[i, n + 1 - i], {n, 1, 12}, {i, 1, n}]]