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.
EXAMPLE
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.
MATHEMATICA
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Clark Kimberling, Nov 23 2019
STATUS
approved