OFFSET
0,2
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.
REFERENCES
L. E. Dickson, History of the Theory of Numbers, vol. 1, Chelsea, New York, 1952, p. 403.
EXAMPLE
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.
MATHEMATICA
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Clark Kimberling, Nov 23 2019
STATUS
approved