A202970 Symmetric matrix based on A001911, by antidiagonals.

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.

Links

Table of n, a(n) for n=1..57.

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

Cf. A202971, A202453, A202876.

Sequence in context: A167787 A185957 A124326 * A205004 A236345 A212991

Adjacent sequences: A202967 A202968 A202969 * A202971 A202972 A202973

Keyword

nonn,tabl

Author

Clark Kimberling, Dec 27 2011

Status

approved