OFFSET
1,2
COMMENTS
Let s=A007598(n+1) (squared Fibonacci numbers, beginning with F(2)), 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 A203003 represents the matrix product M=T'*T. M is the self-fusion matrix of s, as defined at A193722. See A203004 for characteristic polynomials of principal submatrices of M, with interlacing zeros.
EXAMPLE
Northwest corner:
1....4.....9....25....64
4....17....40...109...281
9....40....98...265...685
25...109...265..1865
MATHEMATICA
s[k_] := Fibonacci[k + 1]^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]]; (* A203003 *)
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}] (* A119996 *)
Table[m[1, j], {j, 1, 12}] (* A007598(n+1) *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Dec 27 2011
STATUS
approved