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A202869 Symmetric matrix based on the lower Wythoff sequence, A000201, by antidiagonals.

%I A202869 #9 Oct 24 2024 05:40:24
%S A202869 1,3,3,4,10,4,6,15,15,6,8,22,26,22,8,9,30,39,39,30,9,11,35,54,62,54,
%T A202869 35,11,12,42,66,87,87,66,42,12,14,47,79,108,126,108,79,47,14,16,54,90,
%U A202869 132,159,159,132,90,54,16,17,62,103,151,196,207,196,151,103,62
%N A202869 Symmetric matrix based on the lower Wythoff sequence, A000201, by antidiagonals.
%C A202869 Let s=(1,3,4,6,8,...)=A000201 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 A202869 represents the matrix product M=T'*T. M is the self-fusion matrix of s, as defined at A193722. See A202870 for characteristic polynomials of principal submatrices of M, with interlacing zeros.
%e A202869 Northwest corner:
%e A202869 1...3....4....6....8....9
%e A202869 3...10...15...22...30...35
%e A202869 4...15...26...39...54...66
%e A202869 6...22...39...62...87...108
%e A202869 8...30...54...87...126..159
%t A202869 s[k_] := Floor[k*GoldenRatio];
%t A202869 U = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[s[k], {k, 1, 15}]];
%t A202869 L = Transpose[U]; M = L.U; TableForm[M]
%t A202869 m[i_, j_] := M[[i]][[j]];
%t A202869 Flatten[Table[m[i, n + 1 - i], {n, 1, 12}, {i, 1, n}]]
%t A202869 f[n_] := Sum[m[i, n], {i, 1, n}] + Sum[m[n, j], {j, 1, n - 1}]
%t A202869 Table[f[n], {n, 1, 12}]
%t A202869 Table[Sqrt[f[n]], {n, 1, 12}]   (* A054347 *)
%t A202869 Table[m[1, j], {j, 1, 12}]        (* A000201 *)
%Y A202869 Cf. A202870.
%K A202869 nonn,tabl
%O A202869 1,2
%A A202869 _Clark Kimberling_, Dec 26 2011