A204016 Symmetric matrix based on f(i,j) = max(j mod i, i mod j), by antidiagonals.

0, 1, 1, 1, 0, 1, 1, 2, 2, 1, 1, 2, 0, 2, 1, 1, 2, 3, 3, 2, 1, 1, 2, 3, 0, 3, 2, 1, 1, 2, 3, 4, 4, 3, 2, 1, 1, 2, 3, 4, 0, 4, 3, 2, 1, 1, 2, 3, 4, 5, 5, 4, 3, 2, 1, 1, 2, 3, 4, 5, 0, 5, 4, 3, 2, 1, 1, 2, 3, 4, 5, 6, 6, 5, 4, 3, 2, 1, 1, 2, 3, 4, 5, 6, 0, 6, 5, 4, 3, 2, 1, 1, 2, 3, 4, 5, 6, 7, 7

Offset: 1

Author

Clark Kimberling, Jan 10 2012

Keywords

• nonn
• tabl

Comments

A204016 represents the matrix M given by f(i,j) = max{(j mod i), (i mod j)} for i >= 1 and j >= 1. See A204017 for characteristic polynomials of principal submatrices of M, with interlacing zeros.

Guide to symmetric matrices M based on functions f(i,j) and characteristic polynomial sequences (c.p.s.) with interlaced zeros:

f(i,j)..........................M.........c.p.s.

C(i+j,j)........................A007318...A045912

min(i,j)........................A003983...A202672

max(i,j)........................A051125...A203989

(i+j)*min(i,j)..................A203990...A203991

|i-j|...........................A049581...A203993

max(i-j+1,j-i+1)................A143182...A203992

min(i-j+1,j-i+1)................A203994...A203995

min(i(j+1),j(i+1))..............A203996...A203997

max(i(j+1)-1,j(i+1)-1)..........A203998...A203999

min(i(j+1)-1,j(i+1)-1)..........A204000...A204001

min(2i+j,i+2j)..................A204002...A204003

max(2i+j-2,i+2j-2)..............A204004...A204005

min(2i+j-2,i+2j-2)..............A204006...A204007

max(3i+j-3,i+3j-3)..............A204008...A204011

min(3i+j-3,i+3j-3)..............A204012...A204013

min(3i-2,3j-2)..................A204028...A204029

1+min(j mod i, i mod j).........A204014...A204015

max(j mod i, i mod j)...........A204016...A204017

1+max(j mod i, i mod j).........A204018...A204019

min(i^2,j^2)....................A106314...A204020

min(2i-1, 2j-1).................A157454...A204021

max(2i-1, 2j-1).................A204022...A204023

min(i(i+1)/2,j(j+1)/2)..........A106255...A204024

gcd(i,j)........................A003989...A204025

gcd(i+1,j+1)....................A204030...A204111

min(F(i+1),F(j+1)),F=A000045....A204026...A204027

gcd(F(i+1),F(j+1)),F=A000045....A204112...A204113

gcd(L(i),L(j)),L=A000032........A204114...A204115

gcd(2^i-1,2^j-2)................A204116...A204117

gcd(prime(i),prime(j))..........A204118...A204119

gcd(prime(i+1),prime(j+1))......A204120...A204121

gcd(2^(i-1),2^(j-1))............A144464...A204122

max(floor(i/j),floor(j/i))......A204123...A204124

min(ceiling(i/j),ceiling(j/i))..A204143...A204144

Delannoy matrix.................A008288...A204135

max(2i-j,2j-i)..................A204154...A204155

-1+max(3i-j,3j-i)...............A204156...A204157

max(3i-2j,3j-2i)................A204158...A204159

floor((i+1)/2)..................A204164...A204165

ceiling((i+1)/2)................A204166...A204167

i+j.............................A003057...A204168

i+j-1...........................A002024...A204169

i*j.............................A003991...A204170

..abbreviation below: AOE means "all 1's except"

AOE f(i,i)=i....................A204125...A204126

AOE f(i,i)=A000045(i+1).........A204127...A204128

AOE f(i,i)=A000032(i)...........A204129...A204130

AOE f(i,i)=2i-1.................A204131...A204132

AOE f(i,i)=2^(i-1)..............A204133...A204134

AOE f(i,i)=3i-2.................A204160...A204161

AOE f(i,i)=floor((i+1)/2).......A204162...A204163

...

Other pairs (M, c.p.s.): (A204171, A204172) to (A204183, A204184)

See A202695 for a guide to choices of symmetric matrix M for which the zeros of the characteristic polynomials are all positive.

Examples

			Northwest corner:
  0 1 1 1 1 1 1 1
  0 1 2 2 2 2 2 2
  1 2 0 3 3 3 3 3
  1 2 3 0 4 4 4 4
  1 2 3 4 0 5 5 5
  1 2 3 4 5 0 6 6
  1 2 3 4 5 6 0 7
		

Crossrefs

Cf. A204017, A202453.

Programs

Mathematica

f[i_, j_] := Max[Mod[i, j], Mod[j, i]];
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, 12}, {i, 1, n}]]  (* A204016 *)
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[%]               (* A204017 *)
TableForm[Table[c[n], {n, 1, 10}]]