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A202876
Symmetric matrix based on A000071, by antidiagonals.
4
1, 2, 2, 4, 5, 4, 7, 10, 10, 7, 12, 18, 21, 18, 12, 20, 31, 38, 38, 31, 20, 33, 52, 66, 70, 66, 52, 33, 54, 86, 111, 122, 122, 111, 86, 54, 88, 141, 184, 206, 214, 206, 184, 141, 88, 143, 230, 302, 342, 362, 362, 342, 302, 230, 143, 232, 374, 493, 562, 602
OFFSET
1,2
COMMENTS
Let s=A000071 (Fibonacci numbers -1), 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 A202876 represents the matrix product M=T'*T. M is the self-fusion matrix of s, as defined at A193722. See A202877 for characteristic polynomials of principal submatrices of M, with interlacing zeros.
EXAMPLE
Northwest corner:
1....2....4....7....12....20
2....5....10...18...31....52
4....10...21...38...66....111
7....18...38...70...122...206
12...31...66...122..214...362
MATHEMATICA
s[k_] := -1 + Fibonacci[k + 2];
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}] (* A001924 *)
Table[m[1, j], {j, 1, 12}] (* A000071 *)
Table[m[j, j], {j, 1, 12}] (* A202462 *)
CROSSREFS
Sequence in context: A255368 A186101 A284722 * A128900 A136099 A210597
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Dec 26 2011
STATUS
approved