OFFSET
0,3
COMMENTS
The next term a(8) has 146 digits.
Proposition: A003010(n) has only a trailing 4.
Proof: the case n = 0 is trivial since a(0) = 4 has only a digit. For n > 0, we need to prove that the terms have at least a trailing 4 and that this is unique. A003010(n) == 4 (mod 10) because A003010(0) = 4 and 4^2 - 2 = 14 == 4 (mod 10). To prove that this is unique, we prove that the tenth digit of A003010(n) is never equal to 4 for n > 0, or equivalently, that it is odd. Considering that (10*h + 4)^2 = 100*h^2 + 80*h + 16 and 80*h + 16 has the tenth digit of the form 8*k + 1 mod 10, it follows that the tenth digit of A003010(n-1)^2 is odd, and therefore, also that of A003010(n-1)^2 - 2 = A003010(n). QED.
LINKS
Stefano Spezia, Table of n, a(n) for n = 0..10
FORMULA
a(n) = (A003010(n) - 4)/10.
a(n) = 10*a(n-1)^2 + 8*a(n-1) + 1 for n > 0.
MATHEMATICA
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Stefano Spezia, Jul 29 2024
STATUS
approved