A204016 -id:A204016 - cp's OEIS frontend

A204017 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of max{j mod i, i mod j} (A204016).

0, -1, -1, 0, 1, 4, 6, 0, -1, -15, -38, -20, 0, 1, 56, 206, 184, 50, 0, -1, -185, -1072, -1357, -630, -105, 0, 1, 204, 5146, 9276, 6060, 1736, 196, 0, -1, 6209, -17334, -58470, -52452, -21102, -4116, -336, 0, 1, -112400, -67682, 293984

Offset: 1

Clark Kimberling, Jan 10 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 A204016 and A202605 for guides to related sequences.

			Top of the array:
 1... -1
-1.... 0.... 1
 4.... 6.... 0... -1
-15.. -38.. -20... 0... 1
 56... 206.. 184.. 50.. 0.. -1
...
The 1st principal submatrix (ps) of A204016 is {{1}} (using Mathematica matrix notation), with p(1)=1-x and zero-set {1}.
...
The 2nd ps is {{0,1},{1,0}}, with p(2)=-1+x^2 and zero-set {-1,1}.
...
The 3rd ps is {{0,1,1},{1,0,2},{1,2,0}}, with p(3)=4+6x-x^3 and zero-set {-2, -0.732...,2.732...}.
...
The 4th ps is {{0,1,1,1},{1,0,2,2},{1,2,0,3},{1,2,0,3}}, with p(4)=-15-38x-20x^2+x^4 and zero-set {-3, -1.714, -0.553, 5.268}.
...
The interlace property is illustrated for the last two zero-sets by this chain:
-3 < -2 < -1.7 < -0.7 < -0.5 < 2.7 < 5.2

(For references regarding interlacing roots, see A202605.)

Cf. A204016, A202605.

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}]]

A204233 Permanent of the n-th principal submatrix of A204016.

Original entry on oeis.org

0, 1, 4, 49, 792, 18953, 610796, 25648641, 1359184384, 88722005809, 6994262098260, 655126226755025, 71915748374032232, 9144536677714434105, 1333394182537641307324, 221002933797466121742433

Offset: 1

Clark Kimberling, Jan 13 2012

nonn

Cf. A204016.

f[i_, j_] := Max[Mod[i, j], Mod[j, i]];
m[n_] := Table[f[i, j],
{i, 1, n}, {j, 1, n}]  (* A204016 *)
Permanent[m_] :=
  With[{a = Array[x, Length[m]]},
   Coefficient[Times @@ (m.a), Times @@ a]];
Table[Permanent[m[n]], {n, 1, 16}]  (* A204233 *)

A204164 Symmetric matrix based on f(i,j) = floor((i+j)/2), by antidiagonals.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Jan 12 2012

A204164 represents the matrix M given by f(i,j) = floor((i+j)/2) for i >= 1 and j >= 1.  See A204165 for characteristic polynomials of principal submatrices of M, with interlacing zeros.  See A204016 for a guide to other choices of M.
k appears 4k-1 times, k > 0. - Boris Putievskiy, Jun 12 2024
Number of numbers of the form 2k^2+k+1 <= n, for k = 0,1,2,... - Wesley Ivan Hurt, Jun 19 2024

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

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.

Cf. A057211, A204165, A204016, A202453, A370655.

f[i_, j_] := Floor[(i + j)/2];
m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
TableForm[m[8]] (* 8 X 8 principal submatrix *)
Flatten[Table[f[i, n + 1 - i], {n, 1, 15}, {i, 1, n}]]  (* this sequence *)
  (* or *)
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}]]
  (* or *)
a[n_] = Ceiling[(Sqrt[8*n + 1] - 1)/4];
Nmax = 21; Table[a[n], {n, 1, Nmax}] (* Boris Putievskiy, Jun 12 2024 *)

from math import isqrt
def A204164(n): return (m:=isqrt(n>>1))+(n>m*((m<<1)+1)) # Chai Wah Wu, Nov 14 2024

a(n) = ceiling((sqrt(8*n+1)-1)/4). - Boris Putievskiy, Jun 12 2024
a(n) = Sum_{k=1..n} [c(k) = c(k-1)+1], where c(n) = floor(sqrt(2n)+1/2) mod 2 = A057211(n) and [] is the Iverson bracket. - Wesley Ivan Hurt, Jun 23 2024
a(n) = m+1 if n>m(2m+1) and a(n) = m otherwise where m = floor(sqrt(n/2)). - Chai Wah Wu, Nov 14 2024

A204154 Symmetric matrix based on f(i,j) = max(2i-j, 2j-i), by antidiagonals.

Original entry on oeis.org

1, 3, 3, 5, 2, 5, 7, 4, 4, 7, 9, 6, 3, 6, 9, 11, 8, 5, 5, 8, 11, 13, 10, 7, 4, 7, 10, 13, 15, 12, 9, 6, 6, 9, 12, 15, 17, 14, 11, 8, 5, 8, 11, 14, 17, 19, 16, 13, 10, 7, 7, 10, 13, 16, 19, 21, 18, 15, 12, 9, 6, 9, 12, 15, 18, 21, 23, 20, 17, 14, 11, 8, 8, 11, 14, 17, 20

Offset: 1

Clark Kimberling, Jan 12 2012

A204154 represents the matrix M given by f(i,j) = max(2i-j, 2j-i) for i >= 1 and j >= 1.  See A204155 for characteristic polynomials of principal submatrices of M, with interlacing zeros. See A204016 for a guide to other choices of M.
From Nathaniel J. Strout, Nov 14 2019: (Start)
The sum of the terms in the n-th "_|" shape is given by the octagonal numbers, A000567. For example,
      5,
      4,
  5,4,3,
  is considered the 3rd such shape.
The sum of the terms in the n-th antidiagonal is the absolute value of the (n+1)-th term of A266085. (End)

			Northwest corner:
  1, 3, 5, 7, 9, ...
  3, 2, 4, 6, 8, ...
  5, 4, 3, 5, 7, ...
  7, 6, 5, 4, 6, ...
  9, 8, 7, 6, 5, ...
  ...

Robert Israel, Table of n, a(n) for n = 1..10011 (first 141 antidiagonals, flattened)

Cf. A204155, A204016, A202453.

seq(seq(max(3*j-m,2*m-3*j),j=1..m-1),m=2..19); # Robert Israel, Dec 03 2017

f[i_, j_] := Max[2 i - j, 2 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, 15}, {i, 1, n}]]  (* A204154 *)
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[%]                 (* A204155 *)
TableForm[Table[c[n], {n, 1, 10}]]

G.f. as array: (1 + x + y - 7*y*x + 2*y*x^2 + 2*y^2*x)*x*y/((1-x*y)*(1-x)^2*(1-y)^2). - Robert Israel, Dec 03 2017

A204021 Triangle read by rows: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of min(2i-1,2j-1) (A157454).

Original entry on oeis.org

1, 1, -1, 2, -4, 1, 4, -12, 9, -1, 8, -32, 40, -16, 1, 16, -80, 140, -100, 25, -1, 32, -192, 432, -448, 210, -36, 1, 64, -448, 1232, -1680, 1176, -392, 49, -1, 128, -1024, 3328, -5632, 5280, -2688, 672, -64, 1, 256, -2304, 8640, -17472, 20592

Offset: 0

Clark Kimberling, Jan 11 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.
a(0)=1 by convention. - Philippe Deléham, Nov 17 2013
The n roots of the n-th polynomial are 1/(1+cos((2*k-1)*Pi/(2*n))) for k = 1..n. See my pdf in the link section for the proof. - Jianing Song, Dec 01 2023

			Top of the triangle:
  1
  1....-1
  2....-4.....1
  4....-12....9....-1
  8....-32....40...-16....1
  16...-80....140..-100...25....-1
  32...-192...432..-448...210...-36....1
  ...
-448=2*(-100)-2*140-(-32). - _Philippe Deléham_, Nov 17 2013

(For references regarding interlacing roots, see A202605.)

Eric Weisstein's World of Mathematics, Morgan-Voyce polynomials
Jianing Song, Roots of the characteristic polynomials

Cf. A157454, A202605, A204016. A085478, A211957.

f[i_, j_] := Min[2 i - 1, 2 j - 1];
m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
TableForm[m[6]] (* 6x6 principal submatrix *)
Flatten[Table[f[i, n + 1 - i],
  {n, 1, 15}, {i, 1, n}]]   (* A157454 *)
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[%]                  (* A204021 *)
TableForm[Table[c[n], {n, 1, 10}]]

From Peter Bala, May 01 2012: (Start)
The triangle appears to be a signed version of the row reverse of A211957.
If true, then for 0 <= k <= n-1, T(n,k) = (-1)^k*n/(n-k)*2^(n-k-1)*binomial(2*n-k-1,k) and Sum_{k = 0..n} T(n,k)*x^(n-k) = 1/2*(-1)^n*(b(2*n,-2*x) + 1)/b(n,-2*x), where b(n,x) := Sum_{k = 0..n} binomial(n+k,2*k)*x^k are the Morgan-Voyce polynomials of A085478.
Conjectural o.g.f.: t*(1-x-x^2*t)/(1-2*t*(1-x)+t^2*x^2) = (1-x)*t + (2-4*x+x^2)*t^2 + .... (End)
T(n,k)=2*T(n-1,k)-2*T(n-1,k-1)-T(n-2,k-2), T(0,0)=T(1,0)=1, T(1,1)=-1, T(n,k)=0 of k<0 or if k>n. - Philippe Deléham, Nov 17 2013

A204158 Symmetric matrix based on f(i,j)=max(3i-2j, 3j-2i), by antidiagonals.

Original entry on oeis.org

1, 4, 4, 7, 2, 7, 10, 5, 5, 10, 13, 8, 3, 8, 13, 16, 11, 6, 6, 11, 16, 19, 14, 9, 4, 9, 14, 19, 22, 17, 12, 7, 7, 12, 17, 22, 25, 20, 15, 10, 5, 10, 15, 20, 25, 28, 23, 18, 13, 8, 8, 13, 18, 23, 28, 31, 26, 21, 16, 11, 6, 11, 16, 21, 26, 31, 34, 29, 24, 19, 14, 9, 9, 14

Offset: 1

Clark Kimberling, Jan 12 2012

A204158 represents the matrix M given by f(i,j)=max(3i-2j, 3j-2i) for i>=1 and j>=1. See A204159 for characteristic polynomials of principal submatrices of M, with interlacing zeros. See A204016 for a guide to other choices of M.

			Northwest corner:
1....4....7....10...13
4....2....5....8....11
7....5....3....6....9
10...8....6....4....7

Cf. A204159, A204016, A202453.

f[i_, j_] := Max[3 i - 2 j, 3 j - 2 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, 15}, {i, 1, n}]]   (* A204158 *)
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[%]                  (* A204159 *)
TableForm[Table[c[n], {n, 1, 10}]]

A204022 Symmetric matrix based on f(i,j) = max(2i-1, 2j-1), by antidiagonals.

Original entry on oeis.org

1, 3, 3, 5, 3, 5, 7, 5, 5, 7, 9, 7, 5, 7, 9, 11, 9, 7, 7, 9, 11, 13, 11, 9, 7, 9, 11, 13, 15, 13, 11, 9, 9, 11, 13, 15, 17, 15, 13, 11, 9, 11, 13, 15, 17, 19, 17, 15, 13, 11, 11, 13, 15, 17, 19, 21, 19, 17, 15, 13, 11, 13, 15, 17, 19, 21, 23, 21, 19, 17, 15, 13, 13, 15, 17, 19, 21, 23

Offset: 1

Clark Kimberling, Jan 11 2012

This sequence represents the matrix M given by f(i,j) = max(2i-1, 2j-1) for i >= 1 and j >= 1. See A204023 for characteristic polynomials of principal submatrices of M, with interlacing zeros. See A204016 for a guide to other choices of M.

			Northwest corner:
  1 3 5 7 9
  3 3 5 7 9
  5 5 5 7 9
  7 7 7 7 9
  9 9 9 9 9

G. C. Greubel, Rows n = 1..100 of triangle, flattened

Cf. A202453, A204016, A204022, A209302.

Flat(List([1..12], n-> List([1..n], k-> Maximum(2*k-1, 2*(n-k)+1) ))); # G. C. Greubel, Jul 23 2019

[[Max(2*k-1, 2*(n-k)+1): k in [1..n]]: n in [1..12]]; // G. C. Greubel, Jul 23 2019

(* First program *)
f[i_, j_] := Max[2 i - 1, 2 j - 1];
m[n_] := Table[f[i, j], {i, n}, {j, n}]
TableForm[m[6]] (* 6 X 6 principal submatrix *)
Flatten[Table[f[i, n + 1 - i],
  {n, 15}, {i, n}]]                (* A204022 *)
p[n_] := CharacteristicPolynomial[m[n], x];
c[n_] := CoefficientList[p[n], x]
TableForm[Flatten[Table[p[n], {n, 10}]]]
Table[c[n], {n, 12}]
Flatten[%]                         (* A204023 *)
TableForm[Table[c[n], {n, 10}]]
(* Second program *)
Table[Max[2*k-1, 2*(n-k)+1], {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Jul 23 2019 *)

{T(n, k) = max(2*k-1, 2*(n-k)+1)};
for(n=1, 12, for(k=1, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Jul 23 2019

from math import isqrt
def A204022(n): return (m:=isqrt(n<<3)+1>>1)+abs(m**2-(n<<1)+1) # Chai Wah Wu, Jun 08 2025

[[max(2*k-1, 2*(n-k)+1) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Jul 23 2019

From Ridouane Oudra, May 27 2019: (Start)
a(n) = t + |t^2-2n+1|, where t = floor(sqrt(2n-1)+1/2).
a(n) = A209302(2n-1).
a(n) = A002024(n) + |A002024(n)^2-2n+1|.
a(n) = t + |t^2-2n+1|, where t = floor(sqrt(2n)+1/2). (End)

A204026 Symmetric matrix based on f(i,j)=min(F(i+1),F(j+1)), where F=A000045 (Fibonacci numbers), by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 3, 2, 1, 1, 2, 3, 3, 2, 1, 1, 2, 3, 5, 3, 2, 1, 1, 2, 3, 5, 5, 3, 2, 1, 1, 2, 3, 5, 8, 5, 3, 2, 1, 1, 2, 3, 5, 8, 8, 5, 3, 2, 1, 1, 2, 3, 5, 8, 13, 8, 5, 3, 2, 1, 1, 2, 3, 5, 8, 13, 13, 8, 5, 3, 2, 1, 1, 2, 3, 5, 8, 13, 21, 13, 8, 5, 3, 2, 1, 1, 2, 3, 5, 8

Offset: 1

Clark Kimberling, Jan 11 2012

A204026 represents the matrix M given by f(i,j)=min(F(i+1),F(j+1)) for i>=1 and j>=1. See A204027 for characteristic polynomials of principal submatrices of M, with interlacing zeros. See A204016 for a guide to other choices of M.

			Northwest corner:
1 1 1 1 1 1
1 2 2 2 2 2
1 2 3 3 3 3
1 2 3 5 5 5
1 2 3 5 8 8
1 2 3 5 8 13

Cf. A204026, A204016, A202453.

f[i_, j_] := Min[Fibonacci[i + 1], Fibonacci[j + 1]]
m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
TableForm[m[6]] (* 6x6 principal submatrix *)
Flatten[Table[f[i, n + 1 - i],
  {n, 1, 15}, {i, 1, n}]]  (* A204026 *)
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[%]                 (* A204027 *)
TableForm[Table[c[n], {n, 1, 10}]]

A204028 Symmetric matrix based on f(i,j)=min(3i-2,3j-2), by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 4, 4, 1, 1, 4, 7, 4, 1, 1, 4, 7, 7, 4, 1, 1, 4, 7, 10, 7, 4, 1, 1, 4, 7, 10, 10, 7, 4, 1, 1, 4, 7, 10, 13, 10, 7, 4, 1, 1, 4, 7, 10, 13, 13, 10, 7, 4, 1, 1, 4, 7, 10, 13, 16, 13, 10, 7, 4, 1, 1, 4, 7, 10, 13, 16, 16, 13, 10, 7, 4, 1, 1, 4, 7, 10, 13, 16, 19, 16

Offset: 1

Clark Kimberling, Jan 11 2012

A204028 represents the matrix M given by f(i,j)=min(3i-2,3j-2) for i>=1 and j>=1. See A204029 for characteristic polynomials of principal submatrices of M, with interlacing zeros. See A204016 for a guide to other choices of M.

			Northwest corner:
1...1...1...1....1....1
1...4...4...4....4....4
1...4...7...7....7....7
1...4...7...10...10...10
1...4...7...10...13...13

Cf. A204029, A204016, A202453.

f[i_, j_] := Min[3 i - 2, 3 j - 2];
m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
TableForm[m[6]] (* 6x6 principal submatrix *)
Flatten[Table[f[i, n + 1 - i],
  {n, 1, 15}, {i, 1, n}]]  (* A204028 *)
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[%]                 (* A204029 *)
TableForm[Table[c[n], {n, 1, 10}]]

A204118 Symmetric matrix based on f(i,j) = gcd(prime(i), prime(j)), by antidiagonals.

Original entry on oeis.org

2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 17, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Clark Kimberling, Jan 11 2012

A204118 represents the matrix M given by f(i,j) = gcd(prime(i), prime(j)) for i >= 1 and j >= 1. See A204119 for characteristic polynomials of principal submatrices of M, with interlacing zeros. See A204016 for a guide to other choices of M.

			Northwest corner:
  2  1  1  1  1
  1  3  1  1  1
  1  1  5  1  1
  1  1  1  7  1
  1  1  1  1 11

Cf. A204119, A204016, A202453.

f[i_, j_] := GCD[Prime[i], Prime[j]];
m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
TableForm[m[8]] (* 8 X 8 principal submatrix *)
Flatten[Table[f[i, n + 1 - i],
  {n, 1, 15}, {i, 1, n}]]  (* A204118 *)
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[%]                 (* A204119 *)
TableForm[Table[c[n], {n, 1, 10}]]

cp's OEIS Frontend

A204017 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of max{j mod i, i mod j} (A204016).

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A204233 Permanent of the n-th principal submatrix of A204016.

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A204164 Symmetric matrix based on f(i,j) = floor((i+j)/2), by antidiagonals.

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A204154 Symmetric matrix based on f(i,j) = max(2i-j, 2j-i), by antidiagonals.

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A204021 Triangle read by rows: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of min(2i-1,2j-1) (A157454).

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A204158 Symmetric matrix based on f(i,j)=max(3i-2j, 3j-2i), by antidiagonals.

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A204022 Symmetric matrix based on f(i,j) = max(2i-1, 2j-1), by antidiagonals.

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Sage

Formula

A204026 Symmetric matrix based on f(i,j)=min(F(i+1),F(j+1)), where F=A000045 (Fibonacci numbers), by antidiagonals.