A202871 - OEIS

%I #9 Oct 24 2024 05:40:28

%S 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 69,65,29,47,105,112,120,120,112,105,47,76,170,181,195,196,195,181,

%U 170,76,123,275,293,315,318,318,315,293,275,123,199,445,474,510,514

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

%C 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 Northwest corner:

%e 1....3....4....7....11...18

%e 3....10...15...25...40...65

%e 4....15...26...43...69...112

%e 7....25...43...75...120..195

%e 11...40...69...120..196..318

%t s[k_] := LucasL[k];

%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}] (* A027961 *)

%t Table[m[1, j], {j, 1, 12}]    (* A000032 *)

%Y Cf. A202872.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Dec 26 2011