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A202871 Symmetric matrix based on the Lucas sequence, A000032, by antidiagonals.

%I A202871 #9 Oct 24 2024 05:40:28
%S A202871 1,3,3,4,10,4,7,15,15,7,11,25,26,25,11,18,40,43,43,40,18,29,65,69,75,
%T A202871 69,65,29,47,105,112,120,120,112,105,47,76,170,181,195,196,195,181,
%U A202871 170,76,123,275,293,315,318,318,315,293,275,123,199,445,474,510,514
%N A202871 Symmetric matrix based on the Lucas sequence, A000032, by antidiagonals.
%C A202871 Let s=(1,3,4,7,11,...)=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 A202871 represents the matrix product M=T'*T. M is the self-fusion matrix of s, as defined at A193722. See A202872 for characteristic polynomials of principal submatrices of M, with interlacing zeros.
%e A202871 Northwest corner:
%e A202871 1....3....4....7....11...18
%e A202871 3....10...15...25...40...65
%e A202871 4....15...26...43...69...112
%e A202871 7....25...43...75...120..195
%e A202871 11...40...69...120..196..318
%t A202871 s[k_] := LucasL[k];
%t A202871 U = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[s[k], {k, 1, 15}]];
%t A202871 L = Transpose[U]; M = L.U; TableForm[M]
%t A202871 m[i_, j_] := M[[i]][[j]];
%t A202871 Flatten[Table[m[i, n + 1 - i], {n, 1, 12}, {i, 1, n}]]
%t A202871 f[n_] := Sum[m[i, n], {i, 1, n}] + Sum[m[n, j], {j, 1, n - 1}]
%t A202871 Table[f[n], {n, 1, 12}]
%t A202871 Table[Sqrt[f[n]], {n, 1, 12}] (* A027961 *)
%t A202871 Table[m[1, j], {j, 1, 12}]    (* A000032 *)
%Y A202871 Cf. A202872.
%K A202871 nonn,tabl
%O A202871 1,2
%A A202871 _Clark Kimberling_, Dec 26 2011