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

1,2

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.

Links

Table of n, a(n) for n=1..60.

Example

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

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 *)

Crossrefs

Cf. A202872.

Sequence in context: A175796 A115284 A202869 * A368224 A144626 A033707

Adjacent sequences: A202868 A202869 A202870 * A202872 A202873 A202874

Keyword

nonn,tabl

Author

Clark Kimberling, Dec 26 2011

Status

approved