OFFSET
0,1
COMMENTS
For n >= 1, W. Sierpiński proves that a(n) is divisible by 21.
For n >= 1, A. Engel shows that a(n) = a(n-1) * A220294(n-1). - Hans Havermann, Mar 07 2015
REFERENCES
Arthur Engel, Problem-Solving Strategies, Springer, 1998, pages 121-122 (E3, said to be a "recent competition problem from the former USSR").
W. Sierpiński, 250 Problems in Elementary Number Theory. New York: American Elsevier, 1970. Problem #123.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..10
K. Zelator, On a theorem on sums of the form 1+2^(2^n)+2^(2^n+1)+...+2^(2^n+m) and a result linking Fermat with Mersenne numbers, arXiv:0806.1514 [math.GM], 2008.
FORMULA
A070969(n) = sqrt(4*a(n) - 3). a(n+1) = a(n) * (1 + a(n) - A070969(n)) = a(n) * (1 + A087046(n+2)) hence a(n) divides a(n+1). - Michael Somos, Dec 10 2012
MATHEMATICA
Table[1+2^(2^n)+2^(2^(n+1)), {n, 0, 7}] (* Harvey P. Dale, Dec 16 2015 *)
PROG
(Maxima) A220161(n):=1 + 2^(2^n) + 2^(2^(n+1))$ makelist(A220161(n), n, 0, 10); /* Martin Ettl, Dec 10 2012 */
(PARI) vector(10, n, n--; 1 + 2^(2^n) + 2^(2^(n+1))) \\ G. C. Greubel, Aug 10 2018
(Magma) [1 + 2^(2^n) + 2^(2^(n+1)): n in [0..10]]; // G. C. Greubel, Aug 10 2018
(Python)
def a(n): return 1 + 2**(2**n) + 2**(2**(n+1))
print([a(n) for n in range(8)]) # Michael S. Branicky, Jul 21 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
Michel Marcus, Dec 06 2012
STATUS
approved