login
A204162
Symmetric matrix based on f(i,j) = (floor((i+1)/2) if i=j and = 1 otherwise), by antidiagonals.
3
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
1,13
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.
EXAMPLE
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
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}]]
PROG
(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
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Jan 12 2012
EXTENSIONS
Definition corrected to match with terms by Antti Karttunen, Nov 06 2018
STATUS
approved