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A202970
Symmetric matrix based on A001911, by antidiagonals.
3
1, 3, 3, 6, 10, 6, 11, 21, 21, 11, 19, 39, 46, 39, 19, 32, 68, 87, 87, 68, 32, 53, 115, 153, 167, 153, 115, 53, 87, 191, 260, 296, 296, 260, 191, 87, 142, 314, 433, 505, 528, 505, 433, 314, 142, 231, 513, 713, 843, 904, 904, 843, 713, 513, 231, 375, 835
OFFSET
1,2
COMMENTS
Let s=A001911 (F(n+3)-2, where F(n)=A000045(n), the Fibonacci numbers), 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 A202970 represents the matrix product M=T'*T. M is the self-fusion matrix of s, as defined at A193722. See A202971 for characteristic polynomials of principal submatrices of M, with interlacing zeros.
EXAMPLE
Northwest corner:
1...3...6....11...19
3...10..21...39...68
6...21..46...87...153
11..39..87...167..296
19..68..153..296..528
MATHEMATICA
s[k_] := -2 + Fibonacci[k + 3];
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}] (* A001891 *)
Table[m[1, j], {j, 1, 12}] (* A001911 *)
Table[m[j, j], {j, 1, 12}]
Table[m[j, j + 1], {j, 1, 12}]
Table[Sum[m[i, n + 1 - i], {i, 1, n}], {n, 1, 12}] (* A001925 *)
CROSSREFS
KEYWORD
nonn,tabl,changed
AUTHOR
Clark Kimberling, Dec 27 2011
STATUS
approved