A203993 -id:A203993 - cp's OEIS frontend

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

Clark Kimberling, Jan 10 2012

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.

			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

Cf. A204017, A202453.

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}]]

A049581 Table T(n,k) = |n-k| read by antidiagonals (n >= 0, k >= 0).

Original entry on oeis.org

0, 1, 1, 2, 0, 2, 3, 1, 1, 3, 4, 2, 0, 2, 4, 5, 3, 1, 1, 3, 5, 6, 4, 2, 0, 2, 4, 6, 7, 5, 3, 1, 1, 3, 5, 7, 8, 6, 4, 2, 0, 2, 4, 6, 8, 9, 7, 5, 3, 1, 1, 3, 5, 7, 9, 10, 8, 6, 4, 2, 0, 2, 4, 6, 8, 10, 11, 9, 7, 5, 3, 1, 1, 3, 5, 7, 9, 11, 12, 10, 8, 6, 4, 2, 0, 2, 4, 6, 8, 10, 12

Offset: 0

N. J. A. Sloane

Commutative non-associative operator with identity 0. T(nx,kx) = x T(n,k). A multiplicative analog is A089913. - Marc LeBrun, Nov 14 2003
For the characteristic polynomial of the n X n matrix M_n with entries M_n(i, j) = |i-j| see A203993. - Wolfdieter Lang, Feb 04 2018
For the determinant of the n X n matrix M_n with entries M_n(i, j) = |i-j| see A085750. - Bernard Schott, May 13 2020
a(n) = 0 iff n = 4 times triangular number (A046092). - Bernard Schott, May 13 2020

			Displayed as a triangle t(n, k):
  n\k   0 1 2 3 4 5 6 7 8 9 10 ...
  0:    0
  1:    1 1
  2:    2 0 2
  3:    3 1 1 3
  4:    4 2 0 2 4
  5:    5 3 1 1 3 5
  6:    6 4 2 0 2 4 6
  7:    7 5 3 1 1 3 5 7
  8:    8 6 4 2 0 2 4 6 8
  9:    9 7 5 3 1 1 3 5 7 9
  10:  10 8 6 4 2 0 2 4 6 8 10
... reformatted by _Wolfdieter Lang_, Feb 04 2018
Displayed as a table:
  0 1 2 3 4 5 6 ...
  1 0 1 2 3 4 5 ...
  2 1 0 1 2 3 4 ...
  3 2 1 0 1 2 3 ...
  4 3 2 1 0 1 2 ...
  5 4 3 2 1 0 1 ...
  6 5 4 3 2 1 0 ...
  ...

Peter Kagey, Rows n = 0..125 of triangle, flattened

Cf. A003989, A003990, A003991, A003056, A004247, A002260, A004736.

Cf. A089913. Apart from signs, same as A114327. A203993.

a := Flat(List([0..12],n->List([0..n],k->Maximum(k,n-k)-Minimum(k,n-k)))); # Muniru A Asiru, Jan 26 2018

[[Abs(n-2*k): k in [0..n]]: n in [0..12]]; // G. C. Greubel, Jun 07 2019

seq(seq(abs(n-2*k),k=0..n),n=0..12); # Robert Israel, Sep 30 2015

Table[Abs[(n-k) -k], {n,0,12}, {k,0,n}]//Flatten (* Michael De Vlieger, Sep 29 2015 *)
Table[Join[Range[n,0,-2],Range[If[EvenQ[n],2,1],n,2]],{n,0,12}]//Flatten (* Harvey P. Dale, Sep 18 2023 *)

a(n) = abs(2*(n+1)-binomial((sqrtint(8*(n+1))+1)\2, 2)-(binomial(1+floor(1/2 + sqrt(2*(n+1))), 2))-1);
vector(100, n , a(n-1)) \\ Altug Alkan, Sep 29 2015

{t(n,k) = abs(n-2*k)}; \\ G. C. Greubel, Jun 07 2019

from math import isqrt
def A049581(n): return abs((k:=n+1<<1)-((m:=isqrt(k))+(k>m*(m+1)))**2-1) # Chai Wah Wu, Nov 09 2024

[[abs(n-2*k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Jun 07 2019

G.f.: (x + y - 4*x*y + x^2*y + x*y^2)/((1-x)^2*(1-y)^2*(1-x*y)) = (x/(1-x)^2 + y/(1-y)^2)/(1-x*y). T(n,0) = T(0,n) = n; T(n+1,k+1) = T(n,k). - Franklin T. Adams-Watters, Feb 06 2006
a(n) = |A002260(n+1)-A004736(n+1)| or a(n) = |((n+1)-t*(t+1)/2) - ((t*t+3*t+4)/2-(n+1))| where t = floor((-1+sqrt(8*(n+1)-7))/2). - Boris Putievskiy, Dec 24 2012; corrected by Altug Alkan, Sep 30 2015
From Robert Israel, Sep 30 2015: (Start)
If b(n) = a(n+1) - 2*a(n) + a(n-1), then for n >= 3 we have
  b(n) = -1 if n = (j^2+5j+4)/2 for some integer j >= 1
  b(n) = -3 if n = (j^2+5j+6)/2 for some integer j >= 0
  b(n) = 4 if n = 2j^2 + 6j + 4 for some integer j >= 0
  b(n) = 2 if n = 2j^2 + 8j + 7 or 2j^2 + 8j + 8 for some integer j >= 0
  b(n) = 0 otherwise. (End)
Triangle t(n,k) = max(k, n-k) - min(k, n-k). - Peter Luschny, Jan 26 2018
Triangle t(n, k) = |n - 2*k| for n >= 0, k = 0..n. See the Maple and Mathematica programs. Hence t(n, k)= t(n, n-k). - Wolfdieter Lang, Feb 04 2018
a(n) = |t^2 - 2*n - 1|, where t = floor(sqrt(2*n+1) + 1/2). - Ridouane Oudra, Jun 07 2019; Dec 11 2020
As a rectangle, T(n,k) = |n-k| = max(n,k) - min(n,k). - Clark Kimberling, May 11 2020

A085750 Determinant of the symmetric n X n matrix A defined by A[i,j] = |i-j| for 1 <= i,j <= n.

Original entry on oeis.org

0, -1, 4, -12, 32, -80, 192, -448, 1024, -2304, 5120, -11264, 24576, -53248, 114688, -245760, 524288, -1114112, 2359296, -4980736, 10485760, -22020096, 46137344, -96468992, 201326592, -419430400, 872415232, -1811939328, 3758096384, -7784628224, 16106127360

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Jul 21 2003

The determinant of the distance matrix of a tree with vertex set {1,2,...,n}. The distance matrix is the n X n matrix in which the (i,j)-term is the number of edges in the unique path from vertex i to vertex j. [The matrix A in the definition is the distance matrix of the path-tree 1-2-...-n.]
Hankel transform of A100071. Also Hankel transform of C(2n-2,n-1)(-1)^(n-1). Inverse binomial transform of -n. - Paul Barry, Jan 11 2007
Pisano period lengths: 1, 1, 3, 1, 20, 3, 42, 1, 9, 20, 55, 3,156, 42, 60, 1,136, 9,171, 20, ... - R. J. Mathar, Aug 10 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Emmanuel Briand, Luis Esquivias, Álvaro Gutiérrez, Adrián Lillo, and Mercedes Rosas, Determinant of the distance matrix of a tree, Proceedings of the 36th Conference on Formal Power Series and Algebraic Combinatorics (Bochum), Séminaire Lotharingien de Combinatoire 91B (2024) Article #29, 12 pp.
R. L. Graham and H. O. Pollak, On the addressing problem for loop switching, Bell System Tech. J., 50, 1971, 2495-2519.
Tanya Khovanova, Recursive Sequences
R. Merris, The distance spectrum of a tree, J. Graph Theory, 14, No. 3, 1990,365-369.
Index entries for linear recurrences with constant coefficients, signature (-4,-4).

Essentially the same as A001787.

Cf. A085807, A100071, A203993, A204249, A278845, A278847.

seq((-1)^(n-1)*(n-1)*2^(n-2), n = 1 .. 31);

Table[-(-1)^n*2^(n - 2)*(n - 1), {n, 1, 50}] (* Vladimir Joseph Stephan Orlovsky, Jun 10 2011 *)
LinearRecurrence[{-4,-4},{0,-1},40] (* Harvey P. Dale, Apr 14 2014 *)
CoefficientList[Series[-x/(1 + 2 x)^2, {x, 0, 40}], x] (* Vincenzo Librandi, Apr 15 2014 *)

a(n) = (-1)^n*(1-n)<<(n-2) \\ Charles R Greathouse IV, Sep 30 2022

a(n) = (-1)^(n+1) * (n-1) * 2^(n-2) = (-1)^(n+1) * A001787(n-1).
G.f.: -x/(1+2x)^2. - Paul Barry, Jan 11 2007
a(n) = -4*a(n-1) - 4*a(n-2); a(1) = 0, a(1) = -1. - Philippe Deléham, Nov 03 2008
E.g.f.: -x*exp(-2*x). - Stefano Spezia, Sep 30 2022

More terms from Philippe Deléham, Nov 16 2008

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A049581 Table T(n,k) = |n-k| read by antidiagonals (n >= 0, k >= 0).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

PARI

Python

Sage

Formula

A085750 Determinant of the symmetric n X n matrix A defined by A[i,j] = |i-j| for 1 <= i,j <= n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions