A204016 -id:A204016 - cp's OEIS frontend

A204123 Symmetric matrix based on f(i,j)=max([i/j],[j/i]), where [ ]=floor, by antidiagonals.

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

Offset: 1

Clark Kimberling, Jan 11 2012

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.

			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

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

Cf. A204124, A204016, A202453.

f[i_, j_] := Max[Floor[i/j], Floor[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}]]  (* A204123 *)
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[%]                 (* A204124 *)
TableForm[Table[c[n], {n, 1, 10}]]

A204125 Symmetric matrix based on f(i,j)=(i if i=j and 1 otherwise), by antidiagonals.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Jan 11 2012

A204125 represents the matrix M given by f(i,j)=max([i/j],[j/i]) for i>=1 and j>=1. See A204126 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 1 1 1 1
1 1 3 1 1 1
1 1 1 4 1 1
1 1 1 1 5 1
1 1 1 1 1 6

Cf. A204126, A204016, A202453.

f[i_, j_] := 1; f[i_, i_] := 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}]]  (* A204125 *)
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[%]                 (* A204126 *)
TableForm[Table[c[n], {n, 1, 10}]]

A204131 Symmetric matrix based on f(i,j)=(2i-1 if i=j and 1 otherwise), by antidiagonals.

Original entry on oeis.org

1, 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, 9, 1, 1, 1, 1, 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, 1, 1, 1, 1, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Clark Kimberling, Jan 11 2012

A204131 represents the matrix M given by f(i,j)=(2i-1 if i=j and 1 otherwise) for i>=1 and j>=1. See A204132 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 3 1 1 1
1 1 5 1 1
1 1 1 7 1
1 1 1 1 9

Cf. A204132, A204016, A202453.

f[i_, j_] := 1; f[i_, i_] := 2*i - 1;
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}]]  (* A204131 *)
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[%]                 (* A204132 *)
TableForm[Table[c[n], {n, 1, 10}]]

A204166 Symmetric matrix based on f(i,j)=ceiling[(i+j)/2], by antidiagonals.

Original entry on oeis.org

1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 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, 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, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8

Offset: 1

Clark Kimberling, Jan 12 2012

A204166 represents the matrix M given by f(i,j)=ceiling[(i+j)/2] for i>=1 and j>=1. See A204167 for characteristic polynomials of principal submatrices of M, with interlacing zeros. See A204016 for a guide to other choices of M.

			Northwest corner:
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

Cf. A204167, A204016, A202453.

f[i_, j_] := Ceiling[(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}]]  (* A204166 *)
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[%]                 (* A204167 *)
TableForm[Table[c[n], {n, 1, 10}]]

A204018 Symmetric matrix based on f(i,j)=1+max(j mod i, i mod j), by antidiagonals.

Original entry on oeis.org

1, 2, 2, 2, 1, 2, 2, 3, 3, 2, 2, 3, 1, 3, 2, 2, 3, 4, 4, 3, 2, 2, 3, 4, 1, 4, 3, 2, 2, 3, 4, 5, 5, 4, 3, 2, 2, 3, 4, 5, 1, 5, 4, 3, 2, 2, 3, 4, 5, 6, 6, 5, 4, 3, 2, 2, 3, 4, 5, 6, 1, 6, 5, 4, 3, 2, 2, 3, 4, 5, 6, 7, 7, 6, 5, 4, 3, 2, 2, 3, 4, 5, 6, 7, 1, 7, 6, 5, 4, 3, 2, 2, 3, 4, 5, 6, 7, 8, 8

Offset: 1

Clark Kimberling, Jan 11 2012

A204018 represents the matrix M given by f(i,j)=max{1+(j mod i), 1+( i mod j)} for i>=1 and j>=1. See A204019 for characteristic polynomials of principal submatrices of M, with interlacing zeros. See A204016 for a guide to other choices of M.

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

Cf. A204019, A204016, A202453.

f[i_, j_] := 1 + Max[Mod[i, j], Mod[j, i]];
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}]]   (* A204018 *)
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[%]                  (* A204019 *)
TableForm[Table[c[n], {n, 1, 10}]]

A204019 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of max{1+j mod i, 1+i mod j} (A204018).

Original entry on oeis.org

1, -1, -3, -2, 1, 8, 14, 3, -1, -21, -64, -40, -4, 1, 40, 266, 280, 90, 5, -1, 125, -930, -1671, -896, -175, -6, 1, -2940, 854, 8600, 7228, 2352, 308, 7, -1, 35035, 37744, -27334, -50164, -24594, -5376, -504, -8, 1, -372400

Offset: 1

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).  The least zero of p(n) is -n.
For n>1, the least zero of p(n) is exactly 1-n; the greatest, for p(1) to p(5) is represented by (1,3,5.701...,9.158...13.392...).
See A202605 and A204016 for guides to related sequences.

			Top of the array:
 1....-1
-3....-2......1
 8.....14.....3....-1
-21...-64....-40...-4...1

(For references regarding interlacing roots, see A202605.)

Cf. A204018, A202605, A204016.

f[i_, j_] := 1 + Max[Mod[i, j], Mod[j, i]];
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}]]   (* A204018 *)
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[%]                  (* A204019 *)
TableForm[Table[c[n], {n, 1, 10}]]

A204023 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of max(2i-1, 2j-1) (A204022).

Original entry on oeis.org

1, -1, -6, -4, 1, 20, 36, 9, -1, -56, -160, -120, -16, 1, 144, 560, 700, 300, 25, -1, -352, -1728, -3024, -2240, -630, -36, 1, 832, 4928, 11088, 11760, 5880, 1176, 49, -1, -1920, -13312, -36608, -50688, -36960

Offset: 1

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.

			Top of the array:
 1....-1
-6....-4.....1
 20....36....9.....-1
-56...-160..-120...-16....1

(For references regarding interlacing roots, see A202605.)

Cf. A204022, A202605, A204016.

f[i_, j_] := Max[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}]]  (* A204022 *)
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[%]                 (* A204023 *)
TableForm[Table[c[n], {n, 1, 10}]]

A204024 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of min(i(i+1)/2, j(j+1)/2) (A106255).

Original entry on oeis.org

1, -1, 2, -4, 1, 6, -16, 10, -1, 24, -76, 70, -20, 1, 120, -428, 496, -224, 35, -1, 720, -2808, 3808, -2260, 588, -56, 1, 5040, -21096, 32152, -23008, 8140, -1344, 84, -1, 40320, -178848, 298688, -245560, 107328, -24772, 2772

Offset: 1

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.

			Top of the array:
1....-1
2....-4....1
6....-16...10...-1
24...-76...70...-20....1

(For references regarding interlacing roots, see A202605.)

Cf. A106255, A202605, A204016.

f[i_, j_] := Min[i (i + 1)/2, j (j + 1)/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}]]     (* A106255 *)
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[%]                    (* A204024 *)
TableForm[Table[c[n], {n, 1, 10}]]

A204027 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of M (as in A204026), given by min(F(i+1),F(j+1)), where F=A000045 (Fibonacci numbers).

Original entry on oeis.org

1, -1, 1, -3, 1, 1, -5, 6, -1, 2, -12, 21, -11, 1, 6, -40, 86, -70, 19, -1, 30, -212, 508, -510, 214, -32, 1, 240, -1756, 4482, -5056, 2646, -614, 53, -1, 3120, -23308, 61748, -74480, 44002, -12764, 1703, -87, 1, 65520, -495708, 1343084

Offset: 1

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.

			Top of the array:
1....-1
1....-3....1
1....-5....6....-1
2....-12...21...-11....1

(For references regarding interlacing roots, see A202605.)

Cf. A204026, A202605, A204016.

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}]]

A204029 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of f(i,j)=min(3i-2,3j-2) (A204028).

Original entry on oeis.org

1, -1, 3, -5, 1, 9, -21, 12, -1, 27, -81, 75, -22, 1, 81, -297, 378, -195, 35, -1, 243, -1053, 1701, -1260, 420, -51, 1, 729, -3645, 7128, -6885, 3402, -798, 70, -1, 2187, -12393, 28431, -33858, 22275, -7938, 1386, -92, 1, 6561

Offset: 1

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.

			Top of the array:
1....-1
3....-5....1
9....-21...12...-1
27...-81...75...-22....-11

(For references regarding interlacing roots, see A202605.)

Cf. A204028, A202605, A204016.

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}]]

cp's OEIS Frontend

A204123 Symmetric matrix based on f(i,j)=max([i/j],[j/i]), where [ ]=floor, by antidiagonals.

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A204125 Symmetric matrix based on f(i,j)=(i if i=j and 1 otherwise), by antidiagonals.

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A204131 Symmetric matrix based on f(i,j)=(2i-1 if i=j and 1 otherwise), by antidiagonals.

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A204166 Symmetric matrix based on f(i,j)=ceiling[(i+j)/2], by antidiagonals.

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A204018 Symmetric matrix based on f(i,j)=1+max(j mod i, i mod j), by antidiagonals.

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A204019 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of max{1+j mod i, 1+i mod j} (A204018).

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A204023 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of max(2i-1, 2j-1) (A204022).

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A204024 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of min(i(i+1)/2, j(j+1)/2) (A106255).

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Mathematica

A204027 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of M (as in A204026), given by min(F(i+1),F(j+1)), where F=A000045 (Fibonacci numbers).

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