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A202874 Symmetric matrix based on (1,2,3,5,8,13,...), by antidiagonals. 3
1, 2, 2, 3, 5, 3, 5, 8, 8, 5, 8, 13, 14, 13, 8, 13, 21, 23, 23, 21, 13, 21, 34, 37, 39, 37, 34, 21, 34, 55, 60, 63, 63, 60, 55, 34, 55, 89, 97, 102, 103, 102, 97, 89, 55, 89, 144, 157, 165, 167, 167, 165, 157, 144, 89, 144, 233, 254, 267, 270, 272, 270, 267, 254 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Let s=(1,2,3,5,8,13,...)=(F(k+1)), where F=A000045, 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 A202874 represents the matrix product M=T'*T.  M is the self-fusion matrix of s, as defined at A193722.  See A202875 for characteristic polynomials of principal submatrices of M, with interlacing zeros.

LINKS

Table of n, a(n) for n=1..64.

EXAMPLE

Northwest corner:

1....2....3....5....8....13

2....5....8....13...21...34

3....8....14...23...37...60

5....13...23...39...63...102

8....21...37...63...102..167

MATHEMATICA

s[k_] := Fibonacci[k + 1];

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}]  (* A001911 *)

Table[m[1, j], {j, 1, 12}]     (* A000045 *)

Table[m[j, j], {j, 1, 12}]     (* A119996 *)

Table[m[j, j + 1], {j, 1, 12}] (* A180664 *)

Table[Sum[m[i, n + 1 - i], {i, 1, n}], {n, 1, 12}]  (* A002940 *)

CROSSREFS

Cf. A202875.

Sequence in context: A316311 A317266 A067330 * A199512 A303969 A304931

Adjacent sequences:  A202871 A202872 A202873 * A202875 A202876 A202877

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Dec 26 2011

STATUS

approved

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Last modified May 25 04:50 EDT 2019. Contains 323539 sequences. (Running on oeis4.)