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A202871
Symmetric matrix based on the Lucas sequence, A000032, by antidiagonals.
3
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.
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
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Dec 26 2011
STATUS
approved