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A202874 Symmetric matrix based on (1,2,3,5,8,13,...), by antidiagonals.

%I A202874 #6 Jul 12 2012 00:39:53
%S A202874 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 A202874 21,34,55,60,63,63,60,55,34,55,89,97,102,103,102,97,89,55,89,144,157,
%U A202874 165,167,167,165,157,144,89,144,233,254,267,270,272,270,267,254
%N A202874 Symmetric matrix based on (1,2,3,5,8,13,...), by antidiagonals.
%C A202874 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 A202874 Northwest corner:
%e A202874 1....2....3....5....8....13
%e A202874 2....5....8....13...21...34
%e A202874 3....8....14...23...37...60
%e A202874 5....13...23...39...63...102
%e A202874 8....21...37...63...102..167
%t A202874 s[k_] := Fibonacci[k + 1];
%t A202874 U = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[s[k], {k, 1, 15}]];
%t A202874 L = Transpose[U]; M = L.U; TableForm[M]
%t A202874 m[i_, j_] := M[[i]][[j]];
%t A202874 Flatten[Table[m[i, n + 1 - i], {n, 1, 12}, {i, 1, n}]]
%t A202874 f[n_] := Sum[m[i, n], {i, 1, n}] + Sum[m[n, j], {j, 1, n - 1}]
%t A202874 Table[f[n], {n, 1, 12}]
%t A202874 Table[Sqrt[f[n]], {n, 1, 12}]  (* A001911 *)
%t A202874 Table[m[1, j], {j, 1, 12}]     (* A000045 *)
%t A202874 Table[m[j, j], {j, 1, 12}]     (* A119996 *)
%t A202874 Table[m[j, j + 1], {j, 1, 12}] (* A180664 *)
%t A202874 Table[Sum[m[i, n + 1 - i], {i, 1, n}], {n, 1, 12}]  (* A002940 *)
%Y A202874 Cf. A202875.
%K A202874 nonn,tabl
%O A202874 1,2
%A A202874 _Clark Kimberling_, Dec 26 2011