A204124 -id:A204124

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A204016

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

+10
74

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,8

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.

LINKS

Table of n, a(n) for n=1..99.

EXAMPLE

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

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}]]

CROSSREFS

Cf. A204017, A202453.

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Jan 10 2012

STATUS

approved

A204123

Symmetric matrix based on f(i,j)=max([i/j],[j/i]), where [ ]=floor, by antidiagonals.

+10
4

1, 2, 2, 3, 1, 3, 4, 1, 1, 4, 5, 2, 1, 2, 5, 6, 2, 1, 1, 2, 6, 7, 3, 1, 1, 1, 3, 7, 8, 3, 2, 1, 1, 2, 3, 8, 9, 4, 2, 1, 1, 1, 2, 4, 9, 10, 4, 2, 1, 1, 1, 1, 2, 4, 10, 11, 5, 3, 2, 1, 1, 1, 2, 3, 5, 11, 12, 5, 3, 2, 1, 1, 1, 1, 2, 3, 5, 12, 13, 6, 3, 2, 1, 1, 1, 1, 1, 2, 3, 6, 13, 14, 6, 4, 2

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

This sequence represents the matrix M given by f(i,j)=max([i/j],[j/i]) for i>=1 and j>=1. See A204124 for characteristic polynomials of principal submatrices of M, with interlacing zeros. See A204016 for a guide to other choices of M.

LINKS

G. C. Greubel, Table of n, a(n) for the first 100 aintidiagonals

EXAMPLE

Northwest corner:

1 2 3 4 5 6

2 1 1 2 2 3

3 1 1 1 1 2

4 2 1 1 1 1

5 2 1 1 1 1

6 3 2 1 1 1