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A202970 Symmetric matrix based on A001911, by antidiagonals.

%I A202970 #10 Oct 24 2024 05:40:36
%S A202970 1,3,3,6,10,6,11,21,21,11,19,39,46,39,19,32,68,87,87,68,32,53,115,153,
%T A202970 167,153,115,53,87,191,260,296,296,260,191,87,142,314,433,505,528,505,
%U A202970 433,314,142,231,513,713,843,904,904,843,713,513,231,375,835
%N A202970 Symmetric matrix based on A001911, by antidiagonals.
%C A202970 Let s=A001911 (F(n+3)-2, where F(n)=A000045(n), the 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 A202970 represents the matrix product M=T'*T. M is the self-fusion matrix of s, as defined at A193722. See A202971 for characteristic polynomials of principal submatrices of M, with interlacing zeros.
%e A202970 Northwest corner:
%e A202970 1...3...6....11...19
%e A202970 3...10..21...39...68
%e A202970 6...21..46...87...153
%e A202970 11..39..87...167..296
%e A202970 19..68..153..296..528
%t A202970 s[k_] := -2 + Fibonacci[k + 3];
%t A202970 U = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[s[k], {k, 1, 15}]];
%t A202970 L = Transpose[U]; M = L.U; TableForm[M]
%t A202970 m[i_, j_] := M[[i]][[j]];
%t A202970 Flatten[Table[m[i, n + 1 - i], {n, 1, 12}, {i, 1, n}]]
%t A202970 f[n_] := Sum[m[i, n], {i, 1, n}] + Sum[m[n, j], {j, 1, n - 1}]
%t A202970 Table[f[n], {n, 1, 12}]
%t A202970 Table[Sqrt[f[n]], {n, 1, 12}]  (* A001891 *)
%t A202970 Table[m[1, j], {j, 1, 12}]     (* A001911 *)
%t A202970 Table[m[j, j], {j, 1, 12}]
%t A202970 Table[m[j, j + 1], {j, 1, 12}]
%t A202970 Table[Sum[m[i, n + 1 - i], {i, 1, n}], {n, 1, 12}]  (* A001925 *)
%Y A202970 Cf. A202971, A202453, A202876.
%K A202970 nonn,tabl
%O A202970 1,2
%A A202970 _Clark Kimberling_, Dec 27 2011