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

1,2

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.

Links

Table of n, a(n) for n=1..56.

Example

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

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}]]

Crossrefs

Cf. A000290, A202671, A193722.

Sequence in context: A133037 A061886 A059815 * A203003 A339427 A319646

Adjacent sequences: A202667 A202668 A202669 * A202671 A202672 A202673

Keyword

nonn,tabl

Author

Clark Kimberling, Dec 22 2011

Status

approved