A202605 -id:A202605 - cp's OEIS frontend

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).

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}]]

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}]]

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}]]

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}]]

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}]]

A204121 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of f(i,j) = gcd(prime(i+1), prime(j+1)) (A204120).

Original entry on oeis.org

3, -1, 14, -8, 1, 92, -68, 15, -1, 968, -816, 230, -26, 1, 12096, -11248, 3740, -564, 39, -1, 199296, -198400, 73544, -13192, 1222, -56, 1, 3679488, -3877632, 1567824, -320304, 36160, -2280, 75, -1, 82607616, -91008000

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:
    3,   -1;
   14,   -8,    1;
   92,  -68,   15,   -1;
  968, -816,  230,  -26,    1;

(For references regarding interlacing roots, see A202605.)

Cf. A204120, A202605, A204016.

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]] (* 8 X 8 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}]]

A204124 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)) (A144464).

Original entry on oeis.org

1, -1, -3, -2, 1, -1, 11, 3, -1, 6, -6, -29, -4, 1, 1, -13, 8, 56, 5, -1, -1, -6, 71, -46, -102, -6, 1, 0, 4, 8, -128, 73, 161, 7, -1, 1, -4, -76, 126, 322, -164, -245, -8, 1, 1, -33, 63, 285, -295, -629, 277, 351, 9, -1, -4, 22, 121, -256, -722, 662

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,  -2,   1;
  -1,  11,   3,  -1;
   6,  -6, -29,  -4,   1;

(For references regarding interlacing roots, see A202605.)

Cf. A204123, A202605, A204016.

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]] (* 8 X 8 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}]]

cp's OEIS Frontend

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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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).

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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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Mathematica

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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Mathematica

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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Mathematica

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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Mathematica

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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