A204016 -id:A204016 - cp's OEIS frontend

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

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}]]

A204111 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of f(i,j) = gcd(i+1, j+1) (A204030).

Original entry on oeis.org

2, -1, 5, -5, 1, 10, -20, 9, -1, 44, -100, 62, -14, 1, 104, -328, 330, -128, 20, -1, 656, -2208, 2476, -1176, 263, -27, 1, 2624, -10144, 13992, -8880, 2804, -452, 35, -1, 15744, -66112, 102384, -75760, 29512, -6336, 744, -44, 1, 67584

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:
   2,   -1;
   5,   -5,    1;
  10,  -20,    9,   -1;
  44, -100,   62,  -14,    1;

(For references regarding interlacing roots, see A202605.)

Cf. A204030, A202605, A204016.

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}]]

A204113 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the matrix at A204112, given by f(i,j) = gcd(F(i+1), F(j+1)), where F=A000045 (Fibonacci numbers).

Original entry on oeis.org

1, -1, 1, -3, 1, 2, -8, 6, -1, 8, -36, 35, -11, 1, 48, -232, 274, -116, 19, -1, 576, -2880, 3620, -1728, 358, -32, 1, 10368, -52992, 70632, -37192, 8906, -1016, 53, -1, 331776, -1716480, 2354112, -1294352, 332812, -42924, 2805

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;
  2,  -8,   6,  -1;
  8, -36,  35, -11,   1;

(For references regarding interlacing roots, see A202605.)

Cf. A204112, A202605, A204016.

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}]]

A204115 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix from A204114, given by gcd(L(i+1), L(j+1)), where L=A000032 (Lucas numbers).

Original entry on oeis.org

1, -1, 2, -4, 1, 6, -16, 8, -1, 36, -108, 69, -15, 1, 360, -1152, 834, -230, 26, -1, 5280, -17696, 14368, -4668, 682, -44, 1, 147840, -506048, 426568, -147856, 24262, -1952, 73, -1, 6800640, -23573888, 20317360

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,    8,   -1;
  36, -108,   69,  -15,    1;

(For references regarding interlacing roots, see A202605.)

Cf. A204114, A202605, A204016.

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}]]

A204117 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of f(i,j) = gcd(2^i-1, 2^j-1) (A204116).

Original entry on oeis.org

1, -1, 2, -4, 1, 12, -28, 11, -1, 144, -360, 182, -26, 1, 4320, -11088, 5940, -984, 57, -1, 233280, -616032, 348768, -64728, 4506, -120, 1, 29393280, -78086592, 44775936, -8554608, 636444, -19740, 247, -1, 7054387200

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;
   12,  -28,   11,   -1;
  144, -360,  182,  -26,    1;

(For references regarding interlacing roots, see A202605.)

Cf. A204116, A202605, A204016.

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}]]

A204119 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of f(i,j) = gcd(prime(i), prime(j)) (A204118).

Original entry on oeis.org

2, -1, 5, -5, 1, 22, -28, 10, -1, 140, -204, 95, -17, 1, 1448, -2272, 1210, -278, 28, -1, 17856, -29680, 17444, -4732, 637, -41, 1, 291456, -504832, 317576, -96040, 15386, -1328, 58, -1, 5338368, -9577728, 6373968

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:
    2,   -1;
    5,   -5,    1;
   22,  -28,   10,   -1;
  140, -204,   95,  -17,    1;

(For references regarding interlacing roots, see A202605.)

Cf. A204118, A202605, A204016.

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}]]

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

cp's OEIS Frontend

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

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A204111 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of f(i,j) = gcd(i+1, j+1) (A204030).

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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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A204113 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the matrix at A204112, given by f(i,j) = gcd(F(i+1), F(j+1)), where F=A000045 (Fibonacci numbers).

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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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A204115 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix from A204114, given by gcd(L(i+1), L(j+1)), where L=A000032 (Lucas numbers).

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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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A204117 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of f(i,j) = gcd(2^i-1, 2^j-1) (A204116).

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A204119 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of f(i,j) = gcd(prime(i), prime(j)) (A204118).

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