A202453 -id:A202453 - cp's OEIS frontend

A203905 Symmetric matrix based on (1,0,1,0,1,0,1,0,...), by antidiagonals.

1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 2, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 3, 2, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 3, 3, 3, 2, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 3, 3, 4, 3, 3, 2, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Clark Kimberling, Jan 08 2012

Let s be the periodic sequence (1,0,1,0,1,0,...) and let T be the infinite square matrix whose n-th row is formed by putting n-1 zeros before the terms of s. Let T' be the transpose of T. Then A203905 represents the matrix product M=T'*T. M is the self-fusion matrix of s, as defined at A193722. See A203906 for characteristic polynomials of principal submatrices of M, with interlacing zeros.

			Northwest corner:
1...0...1...0...1...0...1...0
0...1...0...1...0...1...0...1
1...0...2...0...2...0...2...0
0...1...0...2...0...2...0...2
1...0...2...0...3...0...3...0

Cf. A203906, A202453.

t = {1, 0}; t1 = Flatten[{t, t, t, t, t, t, t, t, t, t}];
s[k_] := t1[[k]];
U = NestList[Most[Prepend[#, 0]] &, #,
Length[#] - 1] &[Table[s[k], {k, 1, 15}]];
L = Transpose[U]; M = L.U; TableForm[M] (* A203905 *)
m[i_, j_] := M[[i]][[j]];
Flatten[Table[m[i, n + 1 - i], {n, 1, 12}, {i, 1, n}]]

A203945 Symmetric matrix based on (1,0,0,1,0,0,1,0,0,...), by antidiagonals.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 1, 0, 0, 2, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 0, 0, 2, 0, 0, 1, 0, 1, 0, 0, 2, 0, 0, 3, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0

Offset: 1

Clark Kimberling, Jan 08 2012

Let s be the periodic sequence (1,0,0,1,0,0,...) and let T be the infinite square matrix whose n-th row is formed by putting n-1 zeros before the terms of s. Let T' be the transpose of T. Then A203945 represents the matrix product M=T'*T. M is the self-fusion matrix of s, as defined at A193722. See A203946 for characteristic polynomials of principal submatrices of M, with interlacing zeros.

			Northwest corner:
1...0...0...1...0...0...1
0...1...0...0...1...0...0
0...0...1...0...0...1...0
1...0...0...2...0...0...2
0...1...0...0...2...0...0

Cf. A203946, A202453.

t = {1, 0, 0}; t1 = Flatten[{t, t, t, t, t, t, t, t}];
s[k_] := t1[[k]];
U = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[
   Table[s[k], {k, 1, 15}]];
L = Transpose[U]; M = L.U; TableForm[M]
m[i_, j_] := M[[i]][[j]];
Flatten[Table[m[i, n + 1 - i], {n, 1, 12}, {i, 1, n}]]

A203947 Symmetric matrix based on (1,0,1,1,0,1,1,0,1,...), by antidiagonals.

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 1, 2, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 3, 1, 0, 1, 0, 1, 2, 1, 1, 2, 1, 0, 1, 1, 1, 2, 3, 2, 1, 1, 1, 1, 0, 1, 3, 1, 1, 3, 1, 0, 1, 0, 1, 2, 1, 2, 4, 2, 1, 2, 1, 0, 1, 1, 1, 2, 3, 2, 2, 3, 2, 1, 1, 1, 1, 0, 1, 3, 1, 2, 5, 2, 1, 3, 1, 0, 1, 0, 1, 2, 1, 2, 4, 2, 2

Offset: 1

Clark Kimberling, Jan 08 2012

Let s be the periodic sequence (1,0,1,1,0,1,...) and let T be the infinite square matrix whose n-th row is formed by putting n-1 zeros before the terms of s. Let T' be the transpose of T. Then A203947 represents the matrix product M=T'*T. M is the self-fusion matrix of s, as defined at A193722. See A203948 for characteristic polynomials of principal submatrices of M, with interlacing zeros.

			Northwest corner:
1 0 1 1 0 1 1 0
0 1 0 1 1 0 1 1
1 0 2 1 1 0 1 1
1 1 1 3 1 2 3 1
0 1 1 1 3 1 2 3
1 0 2 2 1 4 2 2
1 1 1 3 2 2 5 2

Cf. A203948, A202453.

t = {1, 0, 1};
t1 = Flatten[{t, t, t, t, t, t, t, t, t, t}];
s[k_] := t1[[k]];
U = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[
   Table[s[k], {k, 1, 15}]];
L = Transpose[U]; M = L.U; TableForm[M]  (* A203947 *)
m[i_, j_] := M[[i]][[j]];
Flatten[Table[m[i, n + 1 - i], {n, 1, 12}, {i, 1, n}]]

A203953 Symmetric matrix based on (1,2,1,2,1,2,...), by antidiagonals.

Original entry on oeis.org

1, 2, 2, 1, 5, 1, 2, 4, 4, 2, 1, 5, 6, 5, 1, 2, 4, 6, 6, 4, 2, 1, 5, 6, 10, 6, 5, 1, 2, 4, 6, 8, 8, 6, 4, 2, 1, 5, 6, 10, 11, 10, 6, 5, 1, 2, 4, 6, 8, 10, 10, 8, 6, 4, 2, 1, 5, 6, 10, 11, 15, 11, 10, 6, 5, 1, 2, 4, 6, 8, 10, 12, 12, 10, 8, 6, 4, 2, 1, 5, 6, 10, 11, 15, 16, 15, 11, 10

Offset: 1

Clark Kimberling, Jan 08 2012

Let s be the periodic sequence (1,2,1,2,1,2,...) and let T be the infinite square matrix whose n-th row is formed by putting n-1 zeros before the terms of s. Let T' be the transpose of T. Then A203951 represents the matrix product M=T'*T. M is the self-fusion matrix of s, as defined at A193722. See A203954 for characteristic polynomials of principal submatrices of M, with interlacing zeros.

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

Cf. A203954, A202453.

t = {1, 2}; t1 = Flatten[{t, t, t, t, t, t, t, t, t, t}];
s[k_] := t1[[k]];
U = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[
   Table[s[k], {k, 1, 15}]];
L = Transpose[U]; M = L.U; TableForm[M]  (* A203953 *)
m[i_, j_] := M[[i]][[j]];
Flatten[Table[m[i, n + 1 - i], {n, 1, 12}, {i, 1, n}]]

A203955 Symmetric matrix based on (1,2,3,1,2,3,1,2,3...), by antidiagonals.

Original entry on oeis.org

1, 2, 2, 3, 5, 3, 1, 8, 8, 1, 2, 5, 14, 5, 2, 3, 5, 11, 11, 5, 3, 1, 8, 11, 15, 11, 8, 1, 2, 5, 14, 13, 13, 14, 5, 2, 3, 5, 11, 14, 19, 14, 11, 5, 3, 1, 8, 11, 15, 19, 19, 15, 11, 8, 1, 2, 5, 14, 13, 16, 28, 16, 13, 14, 5, 2, 3, 5, 11, 14, 19, 22, 22, 19, 14, 11, 5, 3, 1, 8

Offset: 1

Clark Kimberling, Jan 08 2012

Let s be the periodic sequence (1,2,3,1,2,3,...) and let T be the infinite square matrix whose n-th row is formed by putting n-1 zeros before the terms of s. Let T' be the transpose of T. Then A203955 represents the matrix product M=T'*T. M is the self-fusion matrix of s, as defined at A193722. See A203956 for characteristic polynomials of principal submatrices of M, with interlacing zeros.

			Northwest corner:
1....2....3....1....2....3
2....5....8....5....5....8
3....8....14...11...11...14
1....5....11...15...13...14

Cf. A203956, A202453.

t = {1, 2, 3}; t1 = Flatten[{t, t, t, t, t, t, t, t, t}];
s[k_] := t1[[k]];
U = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[
   Table[s[k], {k, 1, 15}]];
L = Transpose[U]; M = L.U; TableForm[M]  (* A203955 *)
m[i_, j_] := M[[i]][[j]];
Flatten[Table[m[i, n + 1 - i], {n, 1, 12}, {i, 1, n}]]

A203994 Symmetric matrix based on f(i,j) = (i+j)*min{i,j}, by antidiagonals.

Original entry on oeis.org

1, 0, 0, -1, 1, -1, -2, 0, 0, -2, -3, -1, 1, -1, -3, -4, -2, 0, 0, -2, -4, -5, -3, -1, 1, -1, -3, -5, -6, -4, -2, 0, 0, -2, -4, -6, -7, -5, -3, -1, 1, -1, -3, -5, -7, -8, -6, -4, -2, 0, 0, -2, -4, -6, -8, -9, -7, -5, -3, -1, 1, -1, -3, -5, -7, -9

Offset: 1

Clark Kimberling, Jan 09 2012

A203994 represents the matrix M given by f(i,j) = min(i-j+1,j-i+1) for i >= 1 and j >= 1. See A203995 for characteristic polynomials of principal submatrices of M, with interlacing zeros.

			Northwest corner:
   1    0   -1   -2   -3
   0    1    0   -1   -2
  -1    0    1    0   -1
   2   -1    0    1    0

G. C. Greubel, Antidiagonal rows n = 1..100, flattened

Cf. A203995, A202453.

Flat(List([1..15], n-> List([1..n], k-> Minimum(2*k-n, n-2*k+2) ))); # G. C. Greubel, Jul 23 2019

[Min(2*k-n, n-2*k+2): k in [1..n], n in [1..15]]; // G. C. Greubel, Jul 23 2019

(* First program *)
f[i_, j_] := Min[i - j + 1, j - i + 1];
m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
TableForm[m[6]] (* 6 X 6 principal submatrix *)
Flatten[Table[f[i, n + 1 - i],
{n, 1, 12}, {i, 1, n}]]  (* A203994 *)
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[%]    (* A203995 *)
TableForm[Table[c[n], {n, 1, 10}]]
(* Second program *)
Table[Min[2*k-n, n-2*k+2], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Jul 23 2019 *)

for(n=1,15, for(k=1,n, print1(min(2*k-n, n-2*k+2), ", "))) \\ G. C. Greubel, Jul 23 2019

[[min(2*k-n, n-2*k+2) for k in (1..n)] for n in (1..15)] # G. C. Greubel, Jul 23 2019

A203996 Symmetric matrix based on f(i,j)=min{i(j+1),j(i+1)}, by antidiagonals.

Original entry on oeis.org

2, 3, 3, 4, 6, 4, 5, 8, 8, 5, 6, 10, 12, 10, 6, 7, 12, 15, 15, 12, 7, 8, 14, 18, 20, 18, 14, 8, 9, 16, 21, 24, 24, 21, 16, 9, 10, 18, 24, 28, 30, 28, 24, 18, 10, 11, 20, 27, 32, 35, 35, 32, 27, 20, 11, 12, 22, 30, 36, 40, 42, 40, 36, 30, 22, 12, 13, 24, 33, 40, 45, 48

Offset: 1

Clark Kimberling, Jan 09 2012

A203996 represents the matrix M given by f(i,j)=min{i(j+1),j(i+1)} for i>=1 and j>=1. See A203997 for characteristic polynomials of principal submatrices of M, with interlacing zeros.

			Northwest corner:
2...3....4....5....6....7
3...6....8....10...12...14
4...8....12...15...18...21
5...10...15...20...24...28

Cf. A203995, A202453.

f[i_, j_] := Min[i (j + 1), j (i + 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, 12}, {i, 1, n}]]   (* A203996 *)
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[%]      (* A203997 *)
TableForm[Table[c[n], {n, 1, 10}]]

A203998 Symmetric matrix based on f(i,j)=max{i(j+1)-1,j(i+1)-1}, by antidiagonals.

Original entry on oeis.org

1, 3, 3, 5, 5, 5, 7, 8, 8, 7, 9, 11, 11, 11, 9, 11, 14, 15, 15, 14, 11, 13, 17, 19, 19, 19, 17, 13, 15, 20, 23, 24, 24, 23, 20, 15, 17, 23, 27, 29, 29, 29, 27, 23, 17, 19, 26, 31, 34, 35, 35, 34, 31, 26, 19, 21, 29, 35, 39, 41, 41, 41, 39, 35, 29, 21, 23, 32, 39, 44

Offset: 1

Clark Kimberling, Jan 09 2012

A203998 represents the matrix M given by f(i,j)=max{i(j+1)-1,j(i+1)-1}for i>=1 and j>=1. See A203999 for characteristic polynomials of principal submatrices of M, with interlacing zeros.

			Northwest corner:
1...3....5....7....9
3...5....8....11...14
5...8....11...15...19
7...11...15...19...24

Cf. A203999, A202453.

f[i_, j_] := Max[i (j + 1) - 1, j (i + 1) - 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, 12}, {i, 1, n}]]    (* A203998 *)
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[%]   (* A203999 *)
TableForm[Table[c[n], {n, 1, 10}]]

A204000 Symmetric matrix based on f(i,j)=min{i(j+1)-1,j(i+1)-1}, by antidiagonals.

Original entry on oeis.org

1, 2, 2, 3, 5, 3, 4, 7, 7, 4, 5, 9, 11, 9, 5, 6, 11, 14, 14, 11, 6, 7, 13, 17, 19, 17, 13, 7, 8, 15, 20, 23, 23, 20, 15, 8, 9, 17, 23, 27, 29, 27, 23, 17, 9, 10, 19, 26, 31, 34, 34, 31, 26, 19, 10, 11, 21, 29, 35, 39, 41, 39, 35, 29, 21, 11, 12, 23, 32, 39, 44, 47, 47

Offset: 1

Clark Kimberling, Jan 09 2012

A204000 represents the matrix M given by f(i,j)=min{i(j+1)-1,j(i+1)-1}for i>=1 and j>=1. See A204001 for characteristic polynomials of principal submatrices of M, with interlacing zeros.

			Northwest corner:
1...2....3....4....5....6
2...5....7....9....11...13
3...7....11...14...17...20
4...9....14...19...23...27
5...11...17...23...29...34

Cf. A204001, A202453.

f[i_, j_] := Min[i (j + 1) - 1, j (i + 1) - 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, 12}, {i, 1, n}]]    (* A204000 *)
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[%]              (* A204001 *)
TableForm[Table[c[n], {n, 1, 10}]]

A204002 Symmetric matrix based on f(i,j)=min{2i+j,i+2j}, by antidiagonals.

Original entry on oeis.org

3, 4, 4, 5, 6, 5, 6, 7, 7, 6, 7, 8, 9, 8, 7, 8, 9, 10, 10, 9, 8, 9, 10, 11, 12, 11, 10, 9, 10, 11, 12, 13, 13, 12, 11, 10, 11, 12, 13, 14, 15, 14, 13, 12, 11, 12, 13, 14, 15, 16, 16, 15, 14, 13, 12, 13, 14, 15, 16, 17, 18, 17, 16, 15, 14, 13, 14, 15, 16, 17, 18, 19, 19

Offset: 1

Clark Kimberling, Jan 09 2012

A204002 represents the matrix M given by f(i,j)=min{2i+j,i+2j}for i>=1 and j>=1. See A204003 for characteristic polynomials of principal submatrices of M, with interlacing zeros.

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

Cf. A204003, A202453.

f[i_, j_] := Min[2 i + j, 2 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}]]    (* A204002 *)
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[%]                 (* A204003 *)
TableForm[Table[c[n], {n, 1, 10}]]

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A203905 Symmetric matrix based on (1,0,1,0,1,0,1,0,...), by antidiagonals.

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A203945 Symmetric matrix based on (1,0,0,1,0,0,1,0,0,...), by antidiagonals.

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A203947 Symmetric matrix based on (1,0,1,1,0,1,1,0,1,...), by antidiagonals.

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A203953 Symmetric matrix based on (1,2,1,2,1,2,...), by antidiagonals.

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A203955 Symmetric matrix based on (1,2,3,1,2,3,1,2,3...), by antidiagonals.

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A203994 Symmetric matrix based on f(i,j) = (i+j)*min{i,j}, by antidiagonals.

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A203996 Symmetric matrix based on f(i,j)=min{i(j+1),j(i+1)}, by antidiagonals.

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A203998 Symmetric matrix based on f(i,j)=max{i(j+1)-1,j(i+1)-1}, by antidiagonals.

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A204000 Symmetric matrix based on f(i,j)=min{i(j+1)-1,j(i+1)-1}, by antidiagonals.

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