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

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

0,2

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.

References

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

Links

Table of n, a(n) for n=0..36.

Example

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.

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 *)

Crossrefs

Cf. A329429, A329430, A329431, A329432.

Sequence in context: A127898 A078938 A135888 * A258245 A133366 A049458

Adjacent sequences: A329430 A329431 A329432 * A329434 A329435 A329436

Keyword

nonn,tabf

Author

Clark Kimberling, Nov 23 2019

Status

approved