cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A329429 Irregular triangular array, read by rows: row n shows the coefficients of the polynomial p(n,x) defined in Comments.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 5, 8, 8, 4, 1, 26, 80, 144, 168, 138, 80, 32, 8, 1, 677, 4160, 13888, 31776, 54792, 74624, 82432, 74944, 56472, 35296, 18208, 7664, 2580, 672, 128, 16, 1, 458330, 5632640, 36109952, 158572864, 531441232, 1439520512, 3264101376, 6342205824
Offset: 0

Views

Author

Clark Kimberling, Nov 13 2019

Keywords

Comments

Let f(x) = x^2 + 1, u(0,x) = 1, u(n,x) = f(u(n-1,x)), and p(n,x) = u(n,sqrt(x)). Except for the first term, the sequence (p(n,0)) = (1, 1, 5, 26, 677, ...) is found in A003095 and A008318. This is a strong divisibility sequence, as implied by Dickson's record of a statement by J. J. Sylvester proved by W. S. Foster in 1889.

Examples

			Rows 0..4:
   1;
   1,  1;
   2,  2,   1;
   5,  8,   8,   4,   1;
  26, 80, 144, 168, 138, 80, 32, 8, 1.
Rows 0..4, the polynomials u(n,x):
  1,
  1 + x^2,
  2 + 2 x^2 + x^4,
  5 + 8 x^2 + 8 x^4 + 4 x^6 + x^8,
  26 + 80 x^2 + 144 x^4 + 168 x^6 + 138 x^8 + 80 x^10 + 32 x^12 + 8 x^14 + x^16.

References

  • L. E. Dickson, History of the Theory of Numbers, vol. 1, Chelsea, New York, 1952, p. 403.

Crossrefs

Cf. A003095, A008318, A329430, A329431, A329432, A329433.

Programs

  • Mathematica
    f[x_] := x^2 + 1;  u[0, x_] := 1;
u[1, x_] := f[x]; u[n_, x_] := f[u[n - 1, x]]
Column[Table [Expand[u[n, x]], {n, 0, 5}]] (* A329429 polynomials u(n,x) *)
Table[CoefficientList[u[n, Sqrt[x]], x], {n, 0, 7}]  (* A329429 array *)

Formula

p(n,0) = (1, 1, 2, 5, 26, 677, 458330, ...)
p(n,1) = (1, 2, 5, 26, 677, 458330, ...)
p(n,2) = (2, 5, 26, 677, 458330, ...)
p(n,5) = (5, 26, 677, 458330, ...)
p(n,26) = (26, 677, 458330, ...), etc.;
that is, p(n,p(k,0)) = p(n+k-2,0); there are similar identities for other sequences p(n,h).

A329432 Irregular triangular array, read by rows: row n shows the coefficients of the polynomial p(n,x) defined in Comments.

Original entry on oeis.org

1, 1, 2, 3, 8, 8, 19, 96, 224, 256, 128, 723, 7296, 35456, 105472, 208384, 278528, 245760, 131072, 32768, 1045459, 21100032, 209001984, 1339772928, 6194997248, 21845442560, 60641837056, 134967984128, 243130040320, 355391766528, 419950493696, 396881821696
Offset: 0

Views

Author

Clark Kimberling, Nov 23 2019

Keywords

Comments

Let f(x) = 2 x^2 + 1, u(0,x) = 1, u(n,x) = f(u(n-1),x), and p(n,x) = u(n,sqrt(x)). Then the sequence (p(n,0)) = (1, 1, 3, 19, 723, 1045459, ... ) is a strong divisibility sequence, as implied by Dickson's record of a statement by J. J. Sylvester proved by W. S. Foster in 1889.

Examples

			Rows 0..4:
  1;
  1, 2;
  3, 8, 8;
  19, 96, 224, 256, 128;
  723, 7296, 35456, 105472, 208384, 278528, 245760, 131072, 32768.
Rows 0..4, the polynomials u(n,x):
  1,
  1 + 2 x^2
  3 + 8 x^2 + 8 x^4
  19 + 96 x^2 + 224 x^4 + 256 x^6 + 128 x^8
  723 + 7296 x^2 + 35456 x^4 + 105472 x^6 + 208384 x^8 + 278528 x^10 + 245760 x^12 + 131072 x^14 + 32768 x^16.

References

  • L. E. Dickson, History of the Theory of Numbers, vol. 1, Chelsea, New York, 1952, p. 403.

Crossrefs

Cf. A329429, A329430, A329431, A329433.

Programs

  • Mathematica
    f[x_] := 2 x^2 + 1;  u[0, x_] := 1;
u[1, x_] := f[x]; u[n_, x_] := f[u[n - 1, x]]
Column[Table [Expand[u[n, x]], {n, 0, 5}]] (* A329432 polynomials u(n,x) *)
Table[CoefficientList[u[n, Sqrt[x]], x], {n, 0, 5}]  (* A329432 array *)

A329431 Irregular triangular array, read by rows: row n shows the coefficients of the polynomial p(n,x) defined in Comments.

Original entry on oeis.org

1, 2, 1, 6, 4, 1, 38, 48, 28, 8, 1, 1446, 3648, 4432, 3296, 1628, 544, 120, 16, 1, 2090918, 10550016, 26125248, 41867904, 48398416, 42666880, 29610272, 16475584, 7419740, 2711424, 800992, 189248, 35064, 4928, 496, 32, 1, 4371938082726, 44118436709376
Offset: 0

Views

Author

Clark Kimberling, Nov 23 2019

Keywords

Comments

Let f(x) = x^2 + 2, u(0,x) = 1, u(n,x) = f(u(n-1),x), and p(n,x) = u(n,sqrt(x)).
Then the sequence (p(n,0)) = (1,2,6,38,1446, ... ) is a strong divisibility sequence, as implied by Dickson's record of a statement by J. J. Sylvester proved by W. S. Foster in 1889. p(n,0) = A072191(n) for n >= 1.

Examples

			Rows 0..4:
  1;
  2, 1;
  6, 4, 1;
  38, 48, 28, 8, 1;
  1446, 3648, 4432, 3296, 1628, 544, 120, 16, 1.
Rows 0..4, the polynomials u(n,x):
  1;
  2 + x^2;
  6 + 4 x^2 + x^4;
  38 + 48 x^2 + 28 x^4 + 8 x^6 + x^8;
  1446 + 3648 x^2 + 4432 x^4 + 3296 x^6 + 1628 x^8 + 544 x^10 + 120 x^12 + 16 x^14 + x^16.

References

  • L. E. Dickson, History of the Theory of Numbers, vol. 1, Chelsea, New York, 1952, p. 403.

Crossrefs

Cf. A329429, A329430, A329432, A329433.

Programs

  • Mathematica
    f[x_] := x^2 + 2;  u[0, x_] := 1;
u[1, x_] := f[x]; u[n_, x_] := f[u[n - 1, x]]
Column[Table [Expand[u[n, x]], {n, 0, 5}]] (* A329431 polynomials u(n,x) *)
Table[CoefficientList[u[n, Sqrt[x]], x], {n, 0, 5}]  (* A329431 array *)

A329433 Irregular triangular array, read by rows: row n shows the coefficients of the polynomial p(n,x) defined in Comments.

Original entry on oeis.org

1, 3, 1, 12, 6, 1, 147, 144, 60, 12, 1, 21612, 42336, 38376, 20808, 7350, 1728, 264, 24, 1, 467078547, 1829931264, 3451101120, 4148777664, 3552268752, 2294085888, 1154824416, 461895840, 148272828, 38314944, 7942320, 1306800, 167340, 16128, 1104, 48, 1
Offset: 0

Views

Author

Clark Kimberling, Nov 23 2019

Keywords

Comments

Let f(x) = x^2 + 3, u(0,x) = 1, u(n,x) = f(u(n-1),x), and p(n,x) = u(n,sqrt(x)). Then the sequence (p(n,0)) = (1, 3, 12, 147, 21612, 467078547,... ) is a strong divisibility sequence, as implied by Dickson's record of a statement by J. J. Sylvester proved by W. S. Foster in 1889.

Examples

			Rows 0..4:
  1;
  3, 1;
  12, 6, 1;
  147, 144, 60, 12, 1;
  21612, 42336, 38376, 20808, 7350, 1728, 264, 24, 1.
Rows 0..4, the polynomials u(n,x):
  1;
  3 + x^2;
  12 + 6 x^2 + x^4;
  147 + 144 x^2 + 60 x^4 + 12 x^6 + x^8;
  21612 + 42336 x^2 + 38376 x^4 + 20808 x^6 + 7350 x^8 + 1728 x^10 + 264 x^12 + 24 x^14 + x^16.

References

  • L. E. Dickson, History of the Theory of Numbers, vol. 1, Chelsea, New York, 1952, p. 403.

Crossrefs

Cf. A329429, A329430, A329431, A329432.

Programs

  • Mathematica
    f[x_] := x^2 + 3;  u[0, x_] := 1;
u[1, x_] := f[x]; u[n_, x_] := f[u[n - 1, x]]
Column[Table [Expand[u[n, x]], {n, 0, 5}]] (* A329433 polynomials u(n,x) *)
Table[CoefficientList[u[n, Sqrt[x]], x], {n, 0, 5}]  (* A329433 array *)

A329441 Irregular triangular array, read by rows: row n shows the coefficients of the polynomial p(n,x) defined in Comments.

Original entry on oeis.org

1, 3, 2, 21, 24, 8, 885, 2016, 1824, 768, 128, 1566453, 7136640, 14585472, 17427456, 13300224, 6635520, 2113536, 393216, 32768, 4907550002421, 44716844551680, 193253086462464, 525562214510592, 1006302608418816, 1438003249348608, 1586056913289216
Offset: 0

Views

Author

Clark Kimberling, Dec 07 2019

Keywords

Comments

Let f(x) = 2 x^2 + 3, u(0,x) = 1, u(n,x) = f(u(n-1,x)), and p(n,x) = u(n,sqrt(x)). Then the sequence (p(n,0)) = (1, 3, 21, 885, 1566453, 4907550002421, 48168094052524714211722485, ... ) is a strong divisibility sequence, as implied by Dickson's record of a statement by J. J. Sylvester proved by W. S. Foster in 1889.

Examples

			Rows 0..4:
  1;
  3, 2;
  21, 24, 8;
  885, 2016, 1824, 768, 128;
  1566453, 7136640, 14585472, 17427456, 13300224, 6635520, 2113536, 393216, 32768.
Rows 0..4, the polynomials u(n,x):
  1;
  3 + 2 x^2;
  21 + 24 x^2 + 8 x^4;
  885 + 2016 x^2 + 1824 x^4 + 768 x^6 + 128 x^8;
  1566453 + 7136640 x^2 + 14585472 x^4 + 17427456 x^6 + 13300224 x^8 + 6635520 + x^10 + 2113536 x^12 + 393216 x^14 +
  32768 x^16.

References

  • L. E. Dickson, History of the Theory of Numbers, vol. 1, Chelsea, New York, 1952, p. 403.

Crossrefs

Cf. A329429, A329430, A329431, A329432, A329442.

Programs

  • Mathematica
    f[x_] := 2 x^2 + 3;  u[0, x_] := 1;
u[1, x_] := f[x]; u[n_, x_] := f[u[n - 1, x]]
Column[Table [Expand[u[n, x]], {n, 0, 5}]] (* A329441 polynomials u(n,x) *)
Table[CoefficientList[u[n, Sqrt[x]], x], {n, 0, 5}]  (* A329441 array *)

A329442 Irregular triangular array, read by rows: row n shows the coefficients of the polynomial p(n,x) defined in Comments.

Original entry on oeis.org

1, 2, 3, 14, 36, 27, 590, 3024, 6156, 5832, 2187, 1044302, 10704960, 49225968, 132339744, 227246796, 255091680, 182815704, 76527504, 14348907, 3271700001614, 67075266827520, 652229166810816, 3990988066439808, 17193623473530864, 55281675697126272
Offset: 0

Views

Author

Clark Kimberling, Dec 07 2019

Keywords

Comments

Let f(x) = 3 x^2 + 2, u(0,x) = 1, u(n,x) = f(u(n-1,x)), and p(n,x) = u(n,sqrt(x)). Then the sequence (p(n,0)) = (1, 2, 14, 590, 1044302, 3271700001614, ...) is a strong divisibility sequence, as implied by Dickson's record of a statement by J. J. Sylvester proved by W. S. Foster in 1889.

Examples

			Rows 0..4:
  1;
  2, 3;
  14, 36, 27;
  590, 3024, 6156, 5832, 2187;
  1044302, 10704960, 49225968, 132339744, 227246796, 255091680, 182815704, 76527504, 14348907.
Rows 0..4, the polynomials u(n,x):
  1;
  2 + 3 x^2;
  14 + 36 x^2 + 27 x^4;
  590 + 3024 x^2 + 6156 x^4 + 5832 x^6 + 2187 x^8;
  1044302 + 10704960 x^2 + 49225968 x^4 + 132339744 x^6 + 227246796 x^8 + 255091680 x^10 + 182815704 x^12 + 76527504
  x^14 + 14348907 x^16.

References

  • L. E. Dickson, History of the Theory of Numbers, vol. 1, Chelsea, New York, 1952, p. 403.

Crossrefs

Cf. A329429, A329430, A329431, A329432, A329441.

Programs

  • Mathematica
    f[x_] := 3 x^2 + 2;  u[0, x_] := 1;
u[1, x_] := f[x]; u[n_, x_] := f[u[n - 1, x]]
Column[Table [Expand[u[n, x]], {n, 0, 5}]] (* A329442 polynomials u(n,x) *)
Table[CoefficientList[u[n, Sqrt[x]], x], {n, 0, 5}]  (* A329442 array *)
Showing 1-6 of 6 results.

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