A202453 -id:A202453 - cp's OEIS frontend

A204006 Symmetric matrix based on f(i,j) = min{2i+j-2,i+2j-2}, by antidiagonals.

1, 2, 2, 3, 4, 3, 4, 5, 5, 4, 5, 6, 7, 6, 5, 6, 7, 8, 8, 7, 6, 7, 8, 9, 10, 9, 8, 7, 8, 9, 10, 11, 11, 10, 9, 8, 9, 10, 11, 12, 13, 12, 11, 10, 9, 10, 11, 12, 13, 14, 14, 13, 12, 11, 10, 11, 12, 13, 14, 15, 16, 15, 14, 13, 12, 11, 12, 13, 14, 15, 16, 17, 17, 16, 15, 14, 13, 12

Offset: 1

Clark Kimberling, Jan 09 2012

A204006 represents the matrix M given by f(i,j) = min{2i+j,i+2j} for i>=1 and j>=1. See A204007 for characteristic polynomials of principal submatrices of M, with interlacing zeros.

			Northwest corner:
1...2...3...4....5....6
2...4...5...6....7....8
3...5...7...8....9....10
4...6...8...10...11...12

Cf. A204007, A202453.

f[i_, j_] := Min[2 i + j - 2, 2 j + i - 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, 12}, {i, 1, n}]]   (* A204006 *)
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[%]                (* A204007 *)
TableForm[Table[c[n], {n, 1, 10}]]

A204012 Symmetric matrix based on f(i,j)=min{3i+j-3,i+3j-3}, by antidiagonals.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Jan 10 2012

A204012 represents the matrix M given by f(i,j)=max{3i+j-3,i+3j-3}for i>=1 and j>=1. See A204013 for characteristic polynomials of principal submatrices of M, with interlacing zeros.

			Northwest corner:
1....2....3....4....5....6
2....5....6....7....8....9
3....6....9....10...11...12
4....7....10...13...14...15

Cf. A204013, A202453.

f[i_, j_] := Min[3 i + j - 3, 3 j + i - 3];
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, 12}, {i, 1, n}]]   (* A204012 *)
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[%]  (* A204013 *)
TableForm[Table[c[n], {n, 1, 10}]]

A204014 Symmetric matrix based by antidiagonals, based on f(i,j)=min{1+(j mod i), 1+( i mod j)}.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Jan 10 2012

A204014 represents the matrix M given by f(i,j)=min{1+(j mod i), 1+( i mod j)} for i>=1 and j>=1. See A204015 for characteristic polynomials of principal submatrices of M, with interlacing zeros.

			Northwest corner:
1 1 1 1 1 1
1 1 2 1 2 1
1 2 1 2 3 1
1 1 2 1 2 3
1 2 3 2 1 2

Cf. A204015, A202453.

f[i_, j_] := Min[1 + Mod[i, j], 1 + 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, 12}, {i, 1, n}]]  (* A204014 *)
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[%]               (* A204015 *)
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}]]

A204030 Symmetric matrix based on f(i,j) = gcd(i+1, j+1), by antidiagonals.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Jan 11 2012

A204030 represents the matrix M given by f(i,j) = gcd(i+1, j+1) for i >= 1 and j >= 1. See A204031 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 2 1 2 1 2 1
  1 3 1 1 3 1 1 3
  2 1 4 1 2 1 4 1
  1 1 1 5 1 1 1 1
  2 3 2 1 6 1 2 3
  1 1 1 1 1 7 1 1

Cf. A204111, A204016, A202453.

f[i_, j_] := GCD[i + 1, j + 1];
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}]]  (* A204030 *)
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[%]                 (* A204111 *)
TableForm[Table[c[n], {n, 1, 10}]]

A204112 Symmetric matrix based on f(i,j) = gcd(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, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 8, 1, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13, 1, 1, 1, 1, 1, 1, 2, 1, 5, 2, 1, 1, 2, 5, 1, 2, 1, 1, 1, 3, 1, 1, 1, 21, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Clark Kimberling, Jan 11 2012

A204112 represents the matrix M given by f(i,j) = gcd(F(i+1), F(j+1)) for i >= 1 and j >= 1. See A204113 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  2  1
  1  1  3  1  1  1
  1  1  1  5  1  1
  1  2  1  1  8  1
  1  1  1  1  1 13

Cf. A204113, A204016, A202453.

u[n_] := Fibonacci[n + 1]
f[i_, j_] := GCD[u[i], u[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}]]    (* A204112 *)
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[%]                   (* A204113 *)
TableForm[Table[c[n], {n, 1, 10}]]

A204114 Symmetric matrix based on f(i,j) = gcd(L(i), L(j)), where L=A000032 (Lucas numbers), by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 7, 1, 3, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 4, 1, 1, 18, 1, 1, 4, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 29, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1

Offset: 1

Clark Kimberling, Jan 11 2012

A204114 represents the matrix M given by f(i,j) = gcd(L(i+1), L(j+1)) for i >= 1 and j >= 1. See A204115 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  4  1  1
  1  1  1  7  1
  1  1  1  1 11

Cf. A204115, A204016, A202453.

u[n_] := LucasL[n]
f[i_, j_] := GCD[u[i], u[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}]]    (* A204114 *)
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[%]                   (* A204115 *)
TableForm[Table[c[n], {n, 1, 10}]]

A204116 Symmetric matrix based on f(i,j) = gcd(2^i-1, 2^j-1), by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 3, 7, 3, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 15, 1, 3, 1, 1, 1, 7, 1, 1, 7, 1, 1, 1, 3, 1, 3, 31, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 7, 15, 1, 63, 1, 15, 7, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 3, 127, 3, 1, 3, 1, 3, 1, 1, 1, 7, 1

Offset: 1

Clark Kimberling, Jan 11 2012

A204116 represents the matrix M given by f(i,j) = gcd(2^i-1, 2^j-1) for i >= 1 and j >= 1. See A204117 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  3  1  3
  1  1  7  1
  1  3  1 15

Cf. A204117, A204016, A202453.

f[i_, j_] := GCD[2^i - 1, 2^j - 1];
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}]]  (* A204116 *)
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[%]                 (* A204117 *)
TableForm[Table[c[n], {n, 1, 10}]]

A204120 Symmetric matrix based on f(i,j) = gcd(prime(i+1),prime(j+1)), by antidiagonals.

Original entry on oeis.org

3, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 7, 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, 13, 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 19, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Clark Kimberling, Jan 11 2012

A204120 represents the matrix M given by f(i,j)=GCD(prime(i+1),prime(j+1)) for i>=1 and j>=1. See A204121 for characteristic polynomials of principal submatrices of M, with interlacing zeros. See A204016 for a guide to other choices of M.

Square array with odd primes (A065091) on main diagonal, and 1 at all other entries; array A204118 without its top row and the leftmost column. - Antti Karttunen, Sep 25 2018

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

Antti Karttunen, Table of n, a(n) for n = 1..65703 (the first 362 antidiagonals of array)

Cf. A065091 (main diagonal), A204118, A204121, A204016, A202453.

f[i_, j_] := GCD[Prime[i + 1], Prime[j + 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}]]    (* A204120 *)
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[%]                   (* A204121 *)
TableForm[Table[c[n], {n, 1, 10}]]

up_to = 65703; \\ = binomial(362+1,2)
A204120sq(row,col) = gcd(prime(1+row),prime(1+col));
A204120list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, if(i++ > up_to, return(v)); v[i] = A204120sq((a-(col-1)),col))); (v); };
v204120 = A204120list(up_to);
A204120(n) = v204120[n]; \\ Antti Karttunen, Sep 25 2018

A204127 Symmetric matrix based on f(i,j)=(F(i+1) if i=j and 1 otherwise), where F=A000045 (Fibonacci numbers), 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, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 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, 21, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Clark Kimberling, Jan 11 2012

A204127 represents the matrix M given by f(i,j)=(F(i+1) if i=j and 1 otherwise) for i>=1 and j>=1. See A204128 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 5 1 1
1 1 1 1 8 1
1 1 1 1 1 13

Cf. A204128, A204016, A202453.

f[i_, j_] := 1; f[i_, i_] := Fibonacci[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}]]  (* A204127 *)
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[%]                 (* A204128 *)
TableForm[Table[c[n], {n, 1, 10}]]

cp's OEIS Frontend

A204006 Symmetric matrix based on f(i,j) = min{2i+j-2,i+2j-2}, by antidiagonals.

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A204012 Symmetric matrix based on f(i,j)=min{3i+j-3,i+3j-3}, by antidiagonals.

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A204014 Symmetric matrix based by antidiagonals, based on f(i,j)=min{1+(j mod i), 1+( i mod j)}.

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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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A204030 Symmetric matrix based on f(i,j) = gcd(i+1, j+1), by antidiagonals.

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A204112 Symmetric matrix based on f(i,j) = gcd(F(i+1), F(j+1)), where F=A000045 (Fibonacci numbers), by antidiagonals.

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A204114 Symmetric matrix based on f(i,j) = gcd(L(i), L(j)), where L=A000032 (Lucas numbers), by antidiagonals.

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A204116 Symmetric matrix based on f(i,j) = gcd(2^i-1, 2^j-1), by antidiagonals.

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A204120 Symmetric matrix based on f(i,j) = gcd(prime(i+1),prime(j+1)), by antidiagonals.

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