A203001 Symmetric matrix based on A007598, by antidiagonals.

1, 1, 1, 4, 2, 4, 9, 5, 5, 9, 25, 13, 18, 13, 25, 64, 34, 41, 41, 34, 64, 169, 89, 113, 99, 113, 89, 169, 441, 233, 290, 266, 266, 290, 233, 441, 1156, 610, 765, 689, 724, 689, 765, 610, 1156, 3025, 1597, 1997, 1811, 1866, 1866, 1811, 1997, 1597, 3025

Offset

1,4

Comments

Let s=A007598 (squared 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 A203001 represents the matrix product M=T'*T. M is the self-fusion matrix of s, as defined at A193722. See A203002 for characteristic polynomials of principal submatrices of M, with interlacing zeros.

Links

Table of n, a(n) for n=1..55.

Example

Northwest corner:
1...1...4....9....25....64
1...2...5....13...34....89
4...5...18...41...113...290
9...13..41...99...266...724

Mathematica

s[k_] := 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}]   (* A001654 *)
Table[m[1, j], {j, 1, 12}]      (* A007598 *)
Table[m[2, j], {j, 1, 12}]      (* A001519 *)
Table[m[j, j], {j, 1, 12}]      (* A005969 *)

Crossrefs

Cf. A203002, A202453.

Sequence in context: A007361 A128136 A048147 * A051666 A011382 A011302

Adjacent sequences: A202998 A202999 A203000 * A203002 A203003 A203004

Keyword

nonn,tabl

Author

Clark Kimberling, Dec 27 2011

Status

approved