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Symmetric matrix based on A007598(n+1), by antidiagonals.
3

%I #6 Jul 12 2012 00:39:54

%S 1,4,4,9,17,9,25,40,40,25,64,109,98,109,64,169,281,265,265,281,169,

%T 441,740,685,723,685,740,441,1156,1933,1802,1865,1865,1802,1933,1156,

%U 3025,5065,4709,4910,4819,4910,4709,5065,3025,7921,13256,12337,12827

%N Symmetric matrix based on A007598(n+1), by antidiagonals.

%C 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.

%e Northwest corner:

%e 1....4.....9....25....64

%e 4....17....40...109...281

%e 9....40....98...265...685

%e 25...109...265..1865

%t s[k_] := Fibonacci[k + 1]^2;

%t U = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[s[k], {k, 1, 15}]];

%t L = Transpose[U]; M = L.U; TableForm[M]

%t m[i_, j_] := M[[i]][[j]]; (* A203003 *)

%t Flatten[Table[m[i, n + 1 - i], {n, 1, 12}, {i, 1, n}]]

%t f[n_] := Sum[m[i, n], {i, 1, n}] + Sum[m[n, j], {j, 1, n - 1}]; Table[f[n], {n, 1, 12}]

%t Table[Sqrt[f[n]], {n, 1, 12}] (* A119996 *)

%t Table[m[1, j], {j, 1, 12}] (* A007598(n+1) *)

%Y Cf. A203004, A203001, A202453.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Dec 27 2011