A329432 - OEIS

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A329432

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

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

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OFFSET

0,3

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.

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..31.

EXAMPLE

Rows 0..4:

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 + 2 x^2