A204162 Symmetric matrix based on f(i,j) = (floor((i+1)/2) if i=j and = 1 otherwise), by antidiagonals.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Author

Clark Kimberling, Jan 12 2012

Keywords

• nonn
• tabl

Comments

A204162 represents the matrix M given by f(i,j) = (floor((i+1)/2) if i=j and 1 otherwise) for i >= 1 and j >= 1. See A204163 for characteristic polynomials of principal submatrices of M, with interlacing zeros. See A204016 for a guide to other choices of M.

Examples

			Northwest corner:
  1 1 1 1 1 1
  1 1 1 1 1 1
  1 1 2 1 1 1
  1 1 1 2 1 1
  1 1 1 1 3 1
  1 1 1 1 1 3
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65703 (the first 362 antidiagonals of array)

Crossrefs

Cf. A204163, A204016, A202453.

Programs

Mathematica

f[i_, j_] := 1; f[i_, i_] := Floor[(i + 1)/2];
m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
TableForm[m[8]] (* 8x8 principal submatrix *)
Flatten[Table[f[i, n + 1 - i],
  {n, 1, 15}, {i, 1, n}]]  (* A204162 *)
p[n_] := CharacteristicPolynomial[m[n], x];
c[n_] := CoefficientList[p[n], x]
TableForm[Flatten[Table[p[n], {n, 1, 10}]]]
Table[c[n], {n, 1, 12}]
Flatten[%]                 (* A204163 *)
TableForm[Table[c[n], {n, 1, 10}]]

PARI

up_to = 65703; \\ = binomial(362+1,2)
A204162sq(row,col) = if(row==col,(row+1)\2,1);
A204162list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, if(i++ > up_to, return(v)); v[i] = A204162sq((a-(col-1)),col))); (v); };
v204162 = A204162list(up_to);
A204162(n) = v204162[n]; \\ Antti Karttunen, Nov 06 2018

Extensions

Definition corrected to match with terms by Antti Karttunen, Nov 06 2018