A203001 Symmetric matrix based on A007598, by antidiagonals.

1, 1, 1, 4, 2, 4, 9, 5, 5, 9, 25, 13, 18, 13, 25, 64, 34, 41, 41, 34, 64, 169, 89, 113, 99, 113, 89, 169, 441, 233, 290, 266, 266, 290, 233, 441, 1156, 610, 765, 689, 724, 689, 765, 610, 1156, 3025, 1597, 1997, 1811, 1866, 1866, 1811, 1997, 1597, 3025

Offset: 1

Author

Clark Kimberling, Dec 27 2011

Keywords

• nonn
• tabl

Comments

Let s=A007598 (squared Fibonacci numbers), 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 A203001 represents the matrix product M=T'*T. M is the self-fusion matrix of s, as defined at A193722. See A203002 for characteristic polynomials of principal submatrices of M, with interlacing zeros.

Examples

			Northwest corner:
1...1...4....9....25....64
1...2...5....13...34....89
4...5...18...41...113...290
9...13..41...99...266...724
		

Crossrefs

Cf. A203002, A202453.

Programs

Mathematica

s[k_] := Fibonacci[k]^2;
U = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[s[k], {k, 1, 15}]];
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}]]
f[n_] := Sum[m[i, n], {i, 1, n}] + Sum[m[n, j], {j, 1, n - 1}]
Table[f[n], {n, 1, 12}]
Table[Sqrt[f[n]], {n, 1, 12}]   (* A001654 *)
Table[m[1, j], {j, 1, 12}]      (* A007598 *)
Table[m[2, j], {j, 1, 12}]      (* A001519 *)
Table[m[j, j], {j, 1, 12}]      (* A005969 *)