OFFSET
0,4
COMMENTS
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.
REFERENCES
L. E. Dickson, History of the Theory of Numbers, vol. 1, Chelsea, New York, 1952, p. 403.
FORMULA
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).
EXAMPLE
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.
MATHEMATICA
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Clark Kimberling, Nov 13 2019
STATUS
approved