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Symmetric matrix based on (1,2,3,5,8,13,...), by antidiagonals.
3

%I #6 Jul 12 2012 00:39:53

%S 1,2,2,3,5,3,5,8,8,5,8,13,14,13,8,13,21,23,23,21,13,21,34,37,39,37,34,

%T 21,34,55,60,63,63,60,55,34,55,89,97,102,103,102,97,89,55,89,144,157,

%U 165,167,167,165,157,144,89,144,233,254,267,270,272,270,267,254

%N Symmetric matrix based on (1,2,3,5,8,13,...), by antidiagonals.

%C Let s=(1,2,3,5,8,13,...)=(F(k+1)), where F=A000045, 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 A202874 represents the matrix product M=T'*T. M is the self-fusion matrix of s, as defined at A193722. See A202875 for characteristic polynomials of principal submatrices of M, with interlacing zeros.

%e Northwest corner:

%e 1....2....3....5....8....13

%e 2....5....8....13...21...34

%e 3....8....14...23...37...60

%e 5....13...23...39...63...102

%e 8....21...37...63...102..167

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

%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]];

%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}]

%t Table[f[n], {n, 1, 12}]

%t Table[Sqrt[f[n]], {n, 1, 12}] (* A001911 *)

%t Table[m[1, j], {j, 1, 12}] (* A000045 *)

%t Table[m[j, j], {j, 1, 12}] (* A119996 *)

%t Table[m[j, j + 1], {j, 1, 12}] (* A180664 *)

%t Table[Sum[m[i, n + 1 - i], {i, 1, n}], {n, 1, 12}] (* A002940 *)

%Y Cf. A202875.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Dec 26 2011