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

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, 69, 65, 29, 47, 105, 112, 120, 120, 112, 105, 47, 76, 170, 181, 195, 196, 195, 181, 170, 76, 123, 275, 293, 315, 318, 318, 315, 293, 275, 123, 199, 445, 474, 510, 514

Offset: 1

Author

Clark Kimberling, Dec 26 2011

Keywords

• nonn
• tabl

Comments

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.

Examples

			Northwest corner:
1....3....4....7....11...18
3....10...15...25...40...65
4....15...26...43...69...112
7....25...43...75...120..195
11...40...69...120..196..318
		

Crossrefs

Cf. A202872.

Programs

Mathematica

s[k_] := LucasL[k];
U = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[s[k], {k, 1, 15}]];
L = Transpose[U]; M = L.U; TableForm[M]
m[i_, j_] := M[[i]][[j]];
Flatten[Table[m[i, n + 1 - i], {n, 1, 12}, {i, 1, n}]]
f[n_] := Sum[m[i, n], {i, 1, n}] + Sum[m[n, j], {j, 1, n - 1}]
Table[f[n], {n, 1, 12}]
Table[Sqrt[f[n]], {n, 1, 12}] (* A027961 *)
Table[m[1, j], {j, 1, 12}]    (* A000032 *)