A204164 -id:A204164 - cp's OEIS frontend

A204165 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of floor[(i+j)/2], as in A204164.

1, -1, 1, -3, 1, -1, -2, 6, -1, 0, 4, 4, -10, 1, 0, 0, -15, -4, 15, -1, 0, 0, 0, 36, 3, -21, 1, 0, 0, 0, 0, -84, 4, 28, -1, 0, 0, 0, 0, 0, 160, -16, -36, 1, 0, 0, 0, 0, 0, 0, -300, 40, 45, -1, 0, 0, 0, 0, 0, 0, 0, 500, -75, -55, 1, 0, 0, 0, 0, 0, 0, 0, 0, -825, 130

Offset: 1

Clark Kimberling, Jan 12 2012

Let p(n)=p(n,x) be the characteristic polynomial of the n-th principal submatrix. The zeros of p(n) are real, and they interlace the zeros of p(n+1). See A202605 and A204016 for guides to related sequences.

			Top of the array:
 1....-1
 1....-3.....1
-1....-2.....6....-1
 0.....4.....4....-10...1

(For references regarding interlacing roots, see A202605.)

Cf. A204164, A202605, A204016.

f[i_, j_] := Floor[(i + j)/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}]]  (* A204164 *)
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[%]                 (* A204165 *)
TableForm[Table[c[n], {n, 1, 10}]]

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

Original entry on oeis.org

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}]]

A210530 T(n,k) = (k + 3n - 2 - (k+n-2)(-1)^(k+n))/2 n, k > 0, read by antidiagonals.

Original entry on oeis.org

1, 2, 3, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22

Offset: 1

Boris Putievskiy, Jan 28 2013

Row T(n,k) for odd n is even numbers sandwiched between n's starts from n and 2*n.
Row T(n,k) for even n is odd numbers sandwiched between n's starts from 2*n-1 and n.
Antidiagonal T(1,k), T(2,k-1), ..., T(k,1) for odd k is 1,2,3,...,k.
Antidiagonal T(1,k), T(2,k-1), ..., T(k,1) for even k is k+1, k+2, ..., 2*k+1.
The main diagonal is A000027.
Diagonal, located above the main diagonal T(1,k), T(2,k+1), T(3,k+2), ... for odd k is A000027.
Diagonal, located above the main diagonal T(1,k), T(2,k+1), T(3,k+2), ... for even k is k, k+3, k+6, ..., A016789, A016777, A008585.
Diagonal, located below the main diagonal T(n,1), T(n+1,2), T(n+2,3), ... for odd n is n,n+1, n+2, ... A000027.
Diagonal, located below the main diagonal T(n,1), T(n+1,2), T(n+2,3), ... for even n is 2*n-1, 2*n+2, 2*n+5, ... A008585, A016777, A016789.
The table contains:
A124625 as row 1,
A114753 as column 1,
A109043 as column 2,
A066104 as column 4.

			The start of the sequence as table:
   1   2   1   4   1   6   1   8   1  10
   3   2   5   2   7   2   9   2  11   2
   3   6   3   8   3  10   3  12   3  14
   7   4   9   4  11   4  13   4  15   4
   5  10   5  12   5  14   5  16   5  18
  11   6  13   6  15   6  17   6  19   6
   7  14   7  16   7  18   7  20   7  22
  15   8  17   8  19   8  21   8  23   8
   9  18   9  20   9  22   9  24   9  26
  19  10  21  10  23  10  25  10  27  10
  ...
The start of the sequence as triangle array read by rows:
   1;
   2,  3;
   1,  2,  3;
   4,  5,  6,  7;
   1,  2,  3,  4,  5;
   6,  7,  8,  9, 10, 11;
   1,  2,  3,  4,  5,  6,  7;
   8,  9, 10, 11, 12, 13, 14, 15;
   1,  2,  3,  4,  5,  6,  7,  8,  9;
  10, 11, 12, 13, 14, 15, 16, 17, 18, 19;
  ...
Row number r contains r numbers.
If r is  odd: 1,2,3,...,r.
If r is even: r, r+1, r+3, ..., 2*r-1.
The start of the sequence as array read by rows, the length of row r is 4*r-1.
First 2*r-1 numbers are from the row number 2*r-1 of triangle array, located above.
Last 2*r numbers are from the row number 2*r of triangle array, located above.
  1,2,3;
  1,2,3,4,5,6,7;
  1,2,3,4,5,6,7,8,9,10,11;
  1,2,3,4,5,6,7,8,9,10,11,12,13,14,15;
  1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19;
  ...
Row number r contains 4*r-1 numbers: 1,2,3,...,4*r-1.

Boris Putievskiy, Rows n = 1..140 of triangle, flattened
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.

Cf. A124625, A114753, A109043, A066104, A000027, A016789, A016777, A008585, A204164.

T[n_, k_] := (k+3n-2-(k+n-2)(-1)^(k+n))/2; Table[T[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Nov 17 2018 *)

T(n,k) = (k+3*n-2-(k+n-2)*(-1)^(k+n))/2; \\ Andrew Howroyd, Jan 11 2018

t=int((math.sqrt(8*n-7)-1)/2)
v=int((t+2)/2)
result=n-v*(2*v-3)-1

As table T(n,k) = (k + 3*n - 2 - (k+n-2)*(-1)^(k+n))/2.
As linear sequence
a(n) = A000027(n) - A204164(n)*(2*A204164(n)-3) - 1.
a(n) = n - v*(2*v-3) - 1, where t = floor((-1 + sqrt(8*n-7))/2) and v = floor((t+2)/2).
G.f. of the table: (y*(- 1 + 3*y^2) + x^2*(2 + 5*y - 2*y^2 - 7*y^3) + x^3*(4 + y - 6*y^2 - y^3) + x*(y + 2*y^2 - y^3))/((- 1 + x)^2*(1 + x)^2*(-1 + y)^2*(1 + y)^2). - Stefano Spezia, Nov 17 2018

A370655 Triangle read by rows where row n is a block of length 4*n-1 which is a permutation of the numbers of its constituents.

Original entry on oeis.org

2, 1, 3, 4, 5, 7, 6, 8, 9, 10, 13, 14, 11, 12, 16, 15, 17, 20, 19, 18, 21, 26, 27, 24, 25, 22, 23, 29, 28, 30, 35, 32, 33, 34, 31, 36, 43, 44, 41, 42, 39, 40, 37, 38, 46, 45, 47, 54, 49, 52, 51, 50, 53, 48, 55

Offset: 1

Boris Putievskiy, Feb 24 2024

Generalization of the Cantor numbering method for two adjacent diagonals. A pair of neighboring diagonals are combined into one block.
The sequence is a self-inverse permutation of natural numbers.
The sequence is an intra-block permutation of integer positive numbers.
The sequence A373498 generates the cyclic group C6 under composition. The elements of C6 are the successive compositions of A373498 with itself:  A374494 = A373498(A373498) = A373498^2, A370655 = A373498^3, A374531 = A373498^4, A374447 = A373498^5. The identity element is A000027 = A373498^6. - Boris Putievskiy, Aug 03 2024

			Triangle begins:
     k = 1   2   3   4   5   6   7   8   9  10  11
  n=1:   2,  1,  3;
  n=2:   4,  5,  7,  6,  8,  9, 10;
  n=3:  13, 14, 11, 12, 16, 15, 17, 20, 19, 18, 21;
Subtracting (n-1)*(2*n-1) from each term is row n is a self-inverse permutation of 1 .. 4*n-1,
  2,1,3,
  1,2,4,3,5,6,7,
  3,4,1,2,6,5,7,10,9,8,11,
  ...
The triangle rows can be arranged as two successive upward antidiagonals in an array:
   2,  3,  7, 10, 16, 21, ...
   1,  5,  9, 12, 18, 23, ...
   4,  8, 11, 19, 22, 34, ...
   6, 14, 20, 25, 33, 40, ...
  13, 17, 24, 32, 39, 51, ...
  15, 27, 35, 42, 52, 61, ...

Boris Putievskiy, Table of n, a(n) for n = 1..9870
Boris Putievskiy, Integer Sequences: Irregular Arrays and Intra-Block Permutations, arXiv:2310.18466 [math.CO], 2023.
Index entries for sequences that are permutations of the natural numbers

Cf. A000027, A004767 (row lengths), A204164, A373498, A374447, A374494, A374531.

Nmax = 21;
a[n_] := Module[{L, R, P, Result}, L = Ceiling[(Sqrt[8*n + 1] - 1)/4];
  R = n - (L - 1)*(2*L - 1);
  P = If[R < 2*L - 1, If[Mod[R, 2] == 1, -R + 2*L - 2, -R + 2*L],
    If[R == 2*L - 1, 2*L,
     If[R == 2*L, R - 1, If[Mod[R, 2] == 1, R, 6*L - R]]]];
  Result = P + (L - 1)*(2*L - 1);
  Result]
Table[a[n], {n, 1, Nmax}]

Linear sequence:
a(n) = P(n) + (L(n)-1)*(2*L(n)-1), where L(n) = ceiling((sqrt(8*n+1)-1)/4),
L(n) = A204164(n),
R(n) = n - (L(n)-1)*(2*L(n)-1),
P(n) = -R(n) + 2*L(n)-2, if R(n) < 2*L(n) - 1 and R(n) mod 2 = 1, P(n) = -R(n) + 2*L(n), if R(n) < 2*L(n) - 1 and R(n) mod 2 = 0, P(n) = 2*L(n), if R(n) = 2*L(n) - 1, P(n) = R(n)-1, if R(n) = 2*L(n), P(n) = R(n), if R(n) > 2*L(n) and R(n) mod 2 = 1, P(n) = 6*L(n) - R(n), if R(n) > 2*L(n) and R(n) mod 2 = 0.
Triangular array T(n,k) for 1 <= k <= 4*n-1 (see Example):
T(n,k) = (n-1)*(2*n-1) + P(n,k), where
P(n,k) = 2*n-k-2 if k < 2*n-1 and k mod 2 = 1,
         2*n-k   if k < 2*n-1 and k mod 2 = 0,
         2*k     if k = 2*n-1,
         k-1     if k = 2*n,
         k       if k > 2*n   and k mod 2 = 1,
         6*n-k   if k > 2*n   and k mod 2 = 0.

A373498 a(a(a(n))) = A370655(n).

Original entry on oeis.org

2, 1, 3, 5, 9, 7, 6, 8, 4, 10, 12, 18, 14, 20, 16, 15, 17, 13, 19, 11, 21, 23, 31, 25, 33, 27, 35, 29, 28, 30, 26, 32, 24, 34, 22, 36, 38, 48, 40, 50, 42, 52, 44, 54, 46, 45, 47, 43, 49, 41, 51, 39, 53, 37, 55

Offset: 1

Boris Putievskiy, Jun 17 2024

Triangle read by rows where row n is a block of length 4*n-1 which is a permutation of the numbers of its constituents.
Generalization of the Cantor numbering method for two adjacent diagonals. A pair of neighboring diagonals are combined into one block.
The sequence is an intra-block permutation of positive integers.
The sequence A373498 generates the cyclic group C6 under composition. The elements of C6 are the successive compositions of A373498 with itself:  A374494 =  A373498(A373498) = A373498^2, A370655 = A373498^3, A374531 = A373498^4, A374447 = A373498^5. The identity element is A000027 = A373498^6.  - Boris Putievskiy, Aug 02 2024

			Triangle begins:
     k = 1   2   3   4   5   6   7   8   9  10  11
  n=1:   2,  1,  3;
  n=2:   5,  9,  7,  6,  8,  4, 10;
  n=3:  12, 18, 14, 20, 16, 15, 17, 13, 19, 11, 21;
The triangle's rows can be arranged as two successive upward antidiagonals in an array:
   2,  3,  7, 10, 16, 21, ...
   1,  9,  4, 20, 11, 35, ...
   5,  8, 14, 19, 27, 34, ...
   6, 18, 13, 33, 24, 52, ...
  12, 17, 25, 32, 42, 51, ...
  15, 31, 26, 50, 41, 73, ...
Subtracting (n-1)*(2*n-1) from each term in row n is a permutation of 1 .. 4*n-1:
   2,1,3,
   2,6,4,3,5,1,7,
   2,8,4,10,6,5,7,3,9,1,11,
   ...
The 3rd power of each permutation is equal to the corresponding permutation in example A370655:
   (2,1,3)^3 = (2,1,3),
   (2,6,4,3,5,1,7)^3 = (1,2,4,3,5,6,7),
   (2,8,4,10,6,5,7,3,9,1,11)^3 = (3,4,1,2,6,5,7,10,9,8,11).

Boris Putievskiy, Table of n, a(n) for n = 1..9870 _Boris Putievskiy_, Aug 02 2024
Boris Putievskiy, Integer Sequences: Irregular Arrays and Intra-Block Permutations, arXiv:2310.18466 [math.CO], 2023.
Index entries for sequences that are permutations of the natural numbers
_Boris Putievskiy_, Aug 02 2024

Cf. A000027, A004767 (row lengths), A204164, A370655, A374447, A374494, A374531.

Nmax=21;
a[n_]:=Module[{L,R,P,Result},L=Ceiling[(Sqrt[8*n+1]-1)/4];
R=n-(L-1)*(2*L-1);
P=Which[R<=2*L-1&&Mod[R,2]==1,R+1,R<=2*L-1&&Mod[R,2]==0,R+2*L,R>2*L-1&&Mod[R,2]==1,R,R>2*L-1&&Mod[R,2]==0,4*L-1-R];
Result=P+(L-1)*(2*L-1);
Result]
Table[a[n],{n,1,Nmax}]
Table[a[a[a[n]]],{n,1,Nmax}] (* A370655 *)

Linear sequence:
a(n) = P(n) + (L(n)-1)*(2*L(n)-1), where L(n) = ceiling((sqrt(8*n+1)-1)/4),
L(n) = A204164(n), R(n) = n - (L(n)-1)*(2*L(n)-1),
P(n) = R(n) + 1 if R(n) <= 2*L(n)-1 and R(n) mod 2 = 1, P(n) = 2*L(n) + R(n) if R(n) <= 2*L(n)-1 and R(n) mod 2 = 0, P(n) = R(n) if R(n) > 2*L(n)-1 and R(n) mod 2 = 1, P(n) = 4*L(n) - R(n) - 1 if R(n) > 2*L(n)-1 and R(n) mod 2 = 0.
Triangular array T(n,k) for 1 <= k <= 4*n-1 (see Example):
T(n,k) = (n-1)*(2*n-1) + P(n,k), where P(n,k) = k + 1 if k <= 2*n-1 and k mod 2 = 1, P(n,k) = 2*n + k if k <= 2*n-1 and k mod 2 = 0,
P(n,k) = k if k > 2*n-1 and k mod 2 = 1, P(n,k) = 4*n - k - 1 if k > 2*n-1 and k mod 2 = 0.

A374447 Inverse permutation to A373498.

Original entry on oeis.org

2, 1, 3, 9, 4, 7, 6, 8, 5, 10, 20, 11, 18, 13, 16, 15, 17, 12, 19, 14, 21, 35, 22, 33, 24, 31, 26, 29, 28, 30, 23, 32, 25, 34, 27, 36, 54, 37, 52, 39, 50, 41, 48, 43, 46, 45, 47, 38, 49, 40, 51, 42, 53, 44, 55

Offset: 1

Boris Putievskiy, Jul 08 2024

Triangle read by rows where row n is a block of length 4*n-1 which is a permutation of the numbers of its constituents.
Generalization of the Cantor numbering method for two adjacent diagonals. A pair of neighboring diagonals are combined into one block.
The sequence is an intra-block permutation of positive integers.
The sequence A373498 generates the cyclic group C6 under composition. The elements of C6 are the successive compositions of A373498 with itself:  A374494 = A373498(A373498) = A373498^2, A370655 = A373498^3, A374531 = A373498^4, A374447 = A373498^5. The identity element is A000027 = A373498^6. - Boris Putievskiy, Aug 03 2024

			Triangle begins:
     k = 1   2   3   4   5   6   7   8   9  10  11
  n=1:   2,  1,  3;
  n=2:   9,  4,  7,  6,  8,  5, 10;
  n=3:  20, 11, 18, 13, 16, 15, 17, 12, 19, 14, 21;
The triangle's rows can be arranged as two successive upward antidiagonals in an array:
    2,  3,  7, 10, 16, 21, ...
    1,  4,  5, 13, 14, 26, ...
    9,  8, 18, 19, 31, 34, ...
    6, 11, 12, 24, 25, 41, ...
   20, 17, 33, 32, 50, 51, ...
   15, 22, 23, 39, 40. 60, ...
Subtracting (n-1)*(2*n-1) from each term in row n is a permutation of 1 .. 4*n-1:
    2,1,3,
    6,1,4,3,5,2,7,
   10,1,8,3,6,5,7,2,9,4,11
   ...
The inverse permutation of each permutation in example A373498 is equal to the corresponding permutation above:
   (2,1,3)^(-1) = (2,1,3),
   (2,6,4,3,5,1,7)^(-1) = (6,1,4,3,5,2,7),
   (2,8,4,10,6,5,7,3,9,1,11)^(-1) = (10,1,8,3,6,5,7,2,9,4,11).
The 5th power of each permutation in example A373498 is equal to the corresponding permutation above:
   (2,1,3)^5 = (2,1,3),
   (2,6,4,3,5,1,7)^5 = (6,1,4,3,5,2,7),
   (2,8,4,10,6,5,7,3,9,1,11)^5 = (10,1,8,3,6,5,7,2,9,4,11).

Boris Putievskiy, Table of n, a(n) for n = 1..9870
Boris Putievskiy, Integer Sequences: Irregular Arrays and Intra-Block Permutations, arXiv:2310.18466 [math.CO], 2023.
Index entries for sequences that are permutations of the natural numbers

Cf. A000027, A004767 (row lengths), A204164, A370655, A373498, A374494, A374531.

Nmax=21;
a[n_]:=Module[{L,R,P,Result},L=Ceiling[(Sqrt[8*n+1]-1)/4];
R=n-(L-1)*(2*L-1); P=Which[R<=2*L&&Mod[R,2]==1,4*L-R-1,R<=2*L&&Mod[R,2]==0,R-1,R>2*L&&Mod[R,2]==1,R,R>2*L&&Mod[R,2]==0,-2*L+R];
Result=P+(L-1)*(2*L-1);
Result]
Table[a[n],{n,1,Nmax}]
Nmax=21;
a[n_]:=Module[{L,R,P,Result},L=Ceiling[(Sqrt[8*n+1]-1)/4];
R=n-(L-1)*(2*L-1); P=Which[R<=2*L-1&&Mod[R,2]==1,R+1,R<=2*L-1&&Mod[R,2]==0,R+2*L,R>2*L-1&&Mod[R,2]==1,R,R>2*L-1&&Mod[R,2]==0,4*L-1-R];
Result=P+(L-1)*(2*L-1);
Result]
Table[a[n],{n,1,Nmax}] (* A373498 *)
Table[a[a[a[a[a[n]]]]],{n,1,Nmax}] (* this sequence *)

Linear sequence:
a(n) = P(n) + (L(n)-1)*(2*L(n)-1), where L(n) = ceiling((sqrt(8*n+1)-1)/4),
L(n) = A204164(n), P(n) = 4*L(n) - R(n) - 1, if R(n) <= 2*L(n) and R(n) mod 2 = 1, P(n) = R(n) - 1, if R(n) <= 2*L(n) and R(n) mod 2 = 0, P(n) = R(n), if R(n) > 2*L(n) and R(n) mod 2 = 1, P(n) = - 2*L(n) + R(n), if R(n) > 2*L(n) and R(n) mod 2 = 0.
a(n) = A373498(A373498(A373498(A373498(A373498(n))))).
Triangular array T(n,k) for 1 <= k <= 4*n-1 (see Example):
T(n,k) = (n-1)*(2*n-1) + P(n,k), where P(n,k) = 4*n - k - 1, if k <= 2*n and k mod 2 = 1, P(n,k) = k-1, if k <= 2*n and k mod 2 = 0, P(n,k) = k, if k >  2*n and k mod 2 = 1, P(n,k) = -2*n + k, if k >  2*n and k mod 2 = 0.

A374494 a(n) = A373498(A373498(n)).

Original entry on oeis.org

1, 2, 3, 9, 4, 6, 7, 8, 5, 10, 18, 13, 20, 11, 15, 16, 17, 14, 19, 12, 21, 31, 26, 33, 24, 35, 22, 28, 29, 30, 27, 32, 25, 34, 23, 36, 48, 43, 50, 41, 52, 39, 54, 37, 45, 46, 47, 44, 49, 42, 51, 40, 53, 38, 55

Offset: 1

Boris Putievskiy, Jul 09 2024

Triangle read by rows where row n is a block of length 4*n-1 which is a permutation of the numbers of its constituents.
Generalization of the Cantor numbering method for two adjacent diagonals. A pair of neighboring diagonals are combined into one block.
The sequence is an intra-block permutation of positive integers.
The sequence A373498 generates the cyclic group C6 under composition. The elements of C6 are the successive compositions of A373498 with itself:  A374494 = A373498(A373498) = A373498^2, A370655 = A373498^3, A374531 = A373498^4, A374447 = A373498^5. The identity element is A000027 = A373498^6.  - Boris Putievskiy, Aug 02 2024

			Triangle begins:
     k = 1   2   3   4   5   6   7   8   9  10  11
  n=1:   1,  2,  3;
  n=2:   9,  4,  6,  7,  8,  5, 10;
  n=3:  18, 13, 20, 11, 15, 16, 17, 14, 19, 12, 21;
The triangle's rows can be arranged as two successive upward antidiagonals in an array:
   1,  3,  6, 10, 15, 21, ...
   2,  4,  5, 11, 12, 22, ...
   9,  8, 20, 19, 35, 34, ...
   7, 13, 14, 24, 25, 39, ...
  18, 17, 33, 32, 52, 51, ...
  16, 26, 27, 41, 42, 60, ...
Subtracting (n-1)*(2*n-1) from each term in row n is a permutation of 1 .. 4*n-1:
   1,2,3,
   6,1,3,4,5,2,7,
   8,3,10,1,5,6,7,4,9,2,11
   ...
The 2nd power of each permutation in example A373498 is equal to the corresponding permutation above:
   (2,1,3)^2 = (1,2,3),
   (2,6,4,3,5,1,7)^2 = (6,1,3,4,5,2,7),
   (2,8,4,10,6,5,7,3,9,1,11)^2 = (8,3,10,1,5,6,7,4,9,2,11).

Boris Putievskiy, Table of n, a(n) for n = 1..9870
Boris Putievskiy, Integer Sequences: Irregular Arrays and Intra-Block Permutations, arXiv:2310.18466 [math.CO], 2023.
Index entries for sequences that are permutations of the natural numbers.

Cf. A000027, A004767 (row lengths), A204164, A370655, A373498, A374447, A374531.

Nmax=21;
a[n_]:=Module[{L,R,P,Result},L=Ceiling[(Sqrt[8*n+1]-1)/4];
R=n-(L-1)*(2*L-1);
P=Which[R<=2*L-1&&Mod[R,2]==1,R+1,R<=2*L-1&&Mod[R,2]==0,R+2*L,R>2*L-1&&Mod[R,2]==1,R,R>2*L-1&&Mod[R,2]==0,4*L-1-R];
Result=P+(L-1)*(2*L-1);
Result]
Table[a[n],{n,1,Nmax}] (* A373498 *)
Table[a[a[n]],{n,1,Nmax}] (* this sequence *)
Nmax = 21;
a[n_] := Module[{L, R, P, Result}, L = Ceiling[(Sqrt[8*n + 1] - 1)/4];
  R = n - (L - 1)*(2*L - 1);
  P = Which[R < 2*L - 1 && Mod[R, 2] == 1, 2*L + R + 1, R < 2*L - 1 && Mod[R, 2] == 0, 2*L - R - 1, R >= 2*L - 1 && Mod[R, 2] == 1, R, R >= 2*L - 1 && Mod[R, 2] == 0, 4*L - R];
  Result = P + (L - 1)*(2*L - 1);
  Result]
Table[a[n], {n, 1, Nmax}]

Linear sequence:
a(n) = P(n) + (L(n)-1)*(2*L(n)-1), where L(n) = ceiling((sqrt(8*n+1)-1)/4),
L(n) = A204164(n), R(n) = n - (L(n)-1)*(2*L(n)-1), P(n) = 2*L(n) + R(n) + 1 if R(n) < 2*L(n)-1 and R(n) mod 2 = 1, P(n) = 2*L(n) - R(n) - 1 if R(n) < 2*L(n)-1 and R(n) mod 2 = 0, P(n) = R(n) if R(n) >= 2*L(n)-1 and R(n) mod 2 = 1, P(n) = 4*L(n) - R(n) if R(n) >= 2*L(n)-1 and R(n) mod 2 = 0.
Triangular array T(n,k) for 1 <= k <= 4*n-1 (see Example):
T(n,k) = (n-1)*(2*n-1) + P(n,k), where P(n,k) = 2*n + k + 1 if k < 2*n-1 and k mod 2 = 1, P(n,k) = 2*n - k - 1 if k < 2*n-1 and k mod 2 = 0, P(n,k) = k if k >= 2*n-1 and k mod 2 = 1, P(n,k) = 4*n - k if k >= 2*n-1 and k mod 2 = 0.

A374531 a(n) = A374494(A374494(n)).

Original entry on oeis.org

1, 2, 3, 5, 9, 6, 7, 8, 4, 10, 14, 20, 12, 18, 15, 16, 17, 11, 19, 13, 21, 27, 35, 25, 33, 23, 31, 28, 29, 30, 22, 32, 24, 34, 26, 36, 44, 54, 42, 52, 40, 50, 38, 48, 45, 46, 47, 37, 49, 39, 51, 41, 53, 43, 55

Offset: 1

Boris Putievskiy, Jul 10 2024

Triangle read by rows where row n is a block of length 4*n-1 which is a permutation of the numbers of its constituents.
Generalization of the Cantor numbering method for two adjacent diagonals. A pair of neighboring diagonals are combined into one block.
The sequence is an intra-block permutation of positive integers.
The sequence A373498 generates the cyclic group C6 under composition. The elements of C6 are the successive compositions of A373498 with itself:  A374494 = A373498(A373498) = A373498^2, A370655 = A373498^3, A374531 = A373498^4, A374447 = A373498^5. The identity element is A000027 = A373498^6.  - Boris Putievskiy, Aug 02 2024

			Triangle begins:
     k = 1   2   3   4   5   6   7   8   9  10  11
  n=1:   1,  2,  3;
  n=2:   5,  9,  6,  7,  8,  4, 10;
  n=3:  14, 20, 12, 18, 15, 16, 17, 11, 19, 13, 21;
The triangle's rows can be arranged as two successive upward antidiagonals in an array:
   1,  3,  6, 10, 15, 21, ...
   2,  9,  4, 18, 13, 31, ...
   5,  8, 12, 19, 23, 34, ...
   7, 20, 11, 33, 24, 50, ...
  14, 17, 25, 32, 40, 51, ...
  16, 35, 22, 52, 39, 73, ...
Subtracting (n-1)*(2*n-1) from each term in row n is a permutation of 1 .. 4*n-1:
   1,2,3,
   2,6,3,4,5,1,7
   4,10,2,8,5,6,7,1,9,3,11
   ...
The 4th power of each permutation in example A373498 is equal to the corresponding permutation above:
   (2,1,3)^4 = (1,2,3),
   (2,6,4,3,5,1,7)^4 = (2,6,3,4,5,1,7),
   (2,8,4,10,6,5,7,3,9,1,11)^4 = (4,10,2,8,5,6,7,1,9,3,11).

Boris Putievskiy, Table of n, a(n) for n = 1..9870
Boris Putievskiy, Integer Sequences: Irregular Arrays and Intra-Block Permutations, arXiv:2310.18466 [math.CO], 2023.
Index entries for sequences that are permutations of the natural numbers.

Cf. A000027, A004767 (row lengths), A204164, A370655, A373498, A374447, A374494.

Nmax=21;
a[n_]:=Module[{L,R,P,Result},L=Ceiling[(Sqrt[8*n+1]-1)/4];
R=n-(L-1)*(2*L-1);
P=Which[R<=2*L-1&&Mod[R,2]==1,R+1,R<=2*L-1&&Mod[R,2]==0,R+2*L,R>2*L-1&&Mod[R,2]==1,R,R>2*L-1&&Mod[R,2]==0,4*L-1-R];
Result=P+(L-1)*(2*L-1);
Result]
Table[a[n],{n,1,Nmax}] (* A373498(n) *)
Table[a[a[a[a[n]]]],{n,1,Nmax}] (* this sequence *)
Nmax=21;
a[n_]:=Module[{L,R,P,Result},L=Ceiling[(Sqrt[8*n+1]-1)/4];
R=n-(L-1)*(2*L-1);
P=Which[R<2*L-1&&Mod[R,2]==1,2*L-R-1,R<2*L-1&&Mod[R,2]==0,4*L-R,R==2*L,R,R>=2*L-1&&Mod[R,2]==1,R,R>=2*L-1&&Mod[R,2]==0,-2*L+R-1];
Result=P+(L-1)*(2*L-1);
Result];
Table[a[n],{n,1,Nmax}]

Linear sequence:
a(n) = P(n) + (L(n)-1)*(2*L(n)-1), where L(n) = ceiling((sqrt(8*n+1)-1)/4),
L(n) = A204164(n), R(n) = n - (L(n)-1)*(2*L(n)-1), P(n) = 2*L(n) - R(n) - 1, if R(n) < 2*L(n) - 1 and R(n) mod 2 = 1, P(n) = 4*L(n) - R(n), if R(n) < 2*L(n) - 1 and R(n) mod 2 = 0, P(n) = R(n), if R(n) = 2*L(n), P(n) = R(n), if R(n) >= 2*L(n) - 1 and R(n) mod 2 = 1, P(n) = - 2*L(n) + R(n) - 1, if R(n) >= 2*L(n) - 1 and R(n) mod 2 = 0.
a(n) = A373498(A373498(A373498(A373498(n)))). Boris Putievskiy, Aug 02 2024
Triangular array T(n,k) for 1 <= k <= 4*n-1 (see Example):
T(n,k) = (n-1)*(2*n-1) + P(n,k), where P(n,k) = 2n - k - 1, if k < 2*n-1 and k mod 2 = 1, P(n,k) = 4n - k, if k < 2*n-1 and k mod 2 = 0, P(n,k) = k, if k = 2*n, P(n,k) = k, if k >= 2*n   and k mod 2 = 1, P(n,k) = - 2n + k - 1,   if k >= 2*n   and k mod 2 = 0.

A210521 Array read by downward antidiagonals: T(n,k) = (n+k-1)(n+k-2) + n + floor((n+k)/2)(1-2*floor((n+k)/2)), for n, k > 0.

Original entry on oeis.org

1, 3, 5, 2, 4, 6, 8, 10, 12, 14, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68

Offset: 1

Boris Putievskiy, Jan 26 2013

Enumeration table T(n,k). The order of the list: T(1,1)=1; for k>0: T(1,2*k+1),T(1,2*k); T(2,2*k),T(2,2*k-1); ... T(2*k,2),T(2*k,1); T(2*k+1,1).
The order of the list is descent stairs from the northeast to southwest: step to the west, step to the south, step to the west and so on. The length of each step is 1 or alternation of elements pair adjacent antidiagonals.
Permutation of the natural numbers.
a(n) is a pairing function: a function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} is the set of integer positive numbers.

			The start of the sequence as a table:
   1,  3,  2,  8,  7,  17,  16,  30,  29, ...
   5,  4, 10,  9, 19,  18,  32,  31,  49, ...
   6, 12, 11, 21, 20,  34,  33,  51,  50, ...
  14, 13, 23, 22, 36,  35,  53,  52,  74, ...
  15, 25, 24, 38, 37,  55,  54,  76,  75, ...
  27, 26, 40, 39, 57,  56,  78,  77, 103, ...
  28, 42, 41, 59, 58,  80,  79, 105, 104, ...
  44, 43, 61, 60, 82,  81, 107, 106, 136, ...
  45, 63, 62, 84, 83, 109, 108, 138, 137, ...
  ...
The start of the sequence as a triangular array read by rows:
   1;
   3,  5;
   2,  4,  6;
   8, 10, 12, 14;
   7,  9, 11, 13, 15;
  17, 19, 21, 23, 25, 27;
  16, 18, 20, 22, 24, 26, 28;
  30, 32, 34, 36, 38, 40, 42, 44;
  29, 31, 33, 35, 37, 39, 41, 43, 45;
  ...
The sequence as array read by rows, the length of row r is 4*r-1. First 2*r-1 numbers are from row 2*r-1 of the triangular array above. Last 2*r numbers are from row 2*r of the triangular array. The start of the sequence:
1,3,5;
2,4,6,8,10,12,14;
7,9,11,13,15,17,19,21,23,25,27;
16,18,20,22,24,26,28,30,32,34,36,38,40,42,44;
29,31,33,35,37,39,41,43,45,47,49,51,53,55,57,59,61,63,65;
...
Row r contains 4*r-1 numbers: 2*r^2-5*r+4, 2*r^2-5*r+6, 2*r^2-5*r+8, ..., r*(2*r+3).
Considered as a triangle, the rows have constant parity.

Boris Putievskiy, Rows n = 1..140 of triangle, flattened
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
Eric W. Weisstein, MathWorld: Pairing functions
Index entries for sequences that are permutations of the natural numbers

Cf. A000027, A204164, the main diagonal is A084849.

T[n_, k_] := (n+k-1)(n+k-2) + 2n + Floor[(n+k)/2](1 - 2 Floor[(n+k)/2]);
Table[T[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Dec 03 2018 *)

t=int((math.sqrt(8*n-7)-1)/2)
v=int((t+2)/2)
result=2*n+v*(1-2*v)

As a table: T(n,k) = (n+k-1)*(n+k-2) + 2*n + floor((n+k)/2)*(1-2*floor((n+k)/2)).
As a linear sequence: a(n) = 2*A000027(n) + A204164(n)*(1-2*A204164(n)).
a(n) = 2*n+v*(1-2*v), where t = floor((-1+sqrt(8*n-7))/2) and v = floor((t+2)/2).
G.f. as a table: (2 - 2*y - 5*y^2 + 6*y^3 + 3*y^4 + x*y*(1 + 3*y-5*y^2 + y^3) + x^2*(- 3 + 7*y + 5*y^2 - 11*y^3 - 6*y^4) - x^3*(- 4 + 5*y + 7*y^2 - 9*y^3 + y^4) + x^4*(1 - y - 4*y^2 + y^3 + 7*y^4))/((- 1 + x)^3*(1 + x)^2*(- 1 + y)^3*(1 + y)^2). - Stefano Spezia, Dec 03 2018

A168258 Triangle read by rows, A101688 * A000012 as infinite lower triangular matrices.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 2, 2, 2, 1, 3, 3, 3, 2, 1, 3, 3, 3, 3, 2, 1, 4, 4, 4, 4, 3, 2, 1, 4, 4, 4, 4, 4, 3, 2, 1, 5, 5, 5, 5, 5, 4, 3, 2, 1, 5, 5, 5, 5, 5, 5, 4, 3, 2, 1, 6, 6, 6, 6, 6, 6, 5, 4, 3, 2, 1, 6, 6, 6, 6, 6, 6, 6, 5, 4, 3, 2, 1

Offset: 1

Gary W. Adamson, Nov 21 2009

Row sums = A001318, general pentagonal numbers: (1, 2, 5, 12, 15, 22, ...).
Eigensequence of the triangle = A168259: (1, 2, 6, 14, 38, 96, 254, 656, ...).
The operation A101688 * A000012 transforms rows of A101688 into sequence terms by taking partial sums from the right of A101688 rows. For example, row 3 of A101688 (0, 0, 1, 1) becomes (2, 2, 2, 1). - Gary W. Adamson, Nov 15 2022

			First few rows of the triangle:
  1;
  1, 1;
  2, 2, 1;
  2, 2, 2, 1;
  3, 3, 3, 2, 1;
  3, 3, 3, 3, 2, 1;
  4, 4, 4, 4, 3, 2, 1;
  4, 4, 4, 4, 4, 3, 2, 1;
  5, 5, 5, 5, 5, 4, 3, 2, 1;
  5, 5, 5, 5, 5, 5, 4, 3, 2, 1;
  6, 6, 6, 6, 6, 6, 5, 4, 3, 2, 1;
  6, 6, 6, 6, 6, 6, 6, 5, 4, 3, 2, 1;
  7, 7, 7, 7, 7, 7, 7, 6, 5, 4, 3, 2, 1;
  7, 7, 7, 7, 7, 7, 7, 7, 6, 5, 4, 3, 2, 1;
  8, 8, 8, 8, 8, 8, 8, 8, 7, 6, 5, 4, 3, 2, 1;
  ...

Cf. A001318, A101688, A000012, A168259, A004736, A204164.

T(n, k) = if(binomial(k, n-k)>0, 1, 0); \\ A101688
lista(nn) = my(ma=matrix(nn+1, nn, n, k, T(n-1, k-1)), mb=matrix(nn, nn, n, k, n>=k)); my(m=ma*mb, list=List()); for (n=1, nn, listput(list, vector(n, k, m[n,k]))); Vec(list); \\ Michel Marcus, Nov 16 2022

Triangle read by rows, A101688 * A000012 as infinite lower triangular matrices.

a(n) = min(A004736, A204164); a(n) = min(j, floor((t+2)/2)), where j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Apr 18 2013

Name corrected by Gary W. Adamson, Nov 15 2022

cp's OEIS Frontend

A204165 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of floor[(i+j)/2], as in A204164.

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

Mathematica

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

A210530 T(n,k) = (k + 3*n - 2 - (k+n-2)*(-1)^(k+n))/2 n, k > 0, read by antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A370655 Triangle read by rows where row n is a block of length 4*n-1 which is a permutation of the numbers of its constituents.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A373498 a(a(a(n))) = A370655(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A374447 Inverse permutation to A373498.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A374494 a(n) = A373498(A373498(n)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A374531 a(n) = A374494(A374494(n)).

Views

Author

A210530 T(n,k) = (k + 3n - 2 - (k+n-2)(-1)^(k+n))/2 n, k > 0, read by antidiagonals.

A210521 Array read by downward antidiagonals: T(n,k) = (n+k-1)(n+k-2) + n + floor((n+k)/2)(1-2*floor((n+k)/2)), for n, k > 0.