OFFSET
0,1
COMMENTS
Through a(n) natural numbers 1, 2, 3, ..., a(n), A007677(n) of those terms are members of the upper level Beatty sequence A000572, while A007676(n) of those terms are in the lower level Beatty sequence A006594. Check: a(4) = 26, which has 7 (= A007677(4)) terms in A000572 (3, 7, 11, 14, 18, 22 and 26), while the remaining 19 (= A007676(4)) are members of the lower level Beatty sequence A006594.
a(n)/A007677(n) converges to (1 + e), as n approaches infinity. Check: a(5)/A007677(5) = 119/32 = 3.71875... where (1 + e) = 3.718281828...
a(n)/A007676(n) converges to (1 + 1/e). Check: a(5)/A007676(5) = 119/87 = 1.3678..., where (1 + 1/e) = 1.367879441...
The cutting sequence for y = (1/e)*x is A085369 (see there for more information).
LINKS
Paolo Xausa, Table of n, a(n) for n = 0..1000
EXAMPLE
Convergents to e are generated from the partial quotients of the continued fraction form of e: [2; 1, 2, 1, 1, 4, 1, 1, 6, ...]. The list of convergents begins 2/1, 3/1, 8/3, 11/4, 19/7, 87/32, 106/39, 193/71, 1264/465, ...
a(n) = sum of numerator and denominator of (n+1)-th convergent, with 2/1 = first convergent.
a(5) = 119 = 87 + 32, where 87/32 is the 6th convergent to e.
MATHEMATICA
Plus @@@ NumeratorDenominator[Convergents[E, 35]] (* Paolo Xausa, Sep 13 2025 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Gary W. Adamson, Jun 26 2003
EXTENSIONS
More terms from Colin Barker, Mar 11 2014
Edited and offset changed to 0 by Paolo Xausa, Sep 13 2025
STATUS
approved