0,4

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.

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

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).

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.

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 *)

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

Sequence in context: A380617 A380616 A380615 * A387609 A326617 A371727

Adjacent sequences: A329426 A329427 A329428 * A329430 A329431 A329432

nonn,tabf

Clark Kimberling, Nov 13 2019

approved