A085369 Cutting sequence for 1/e.

1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1

Offset

1,1

Comments

Given the line y = (1/e)*x starting from (0,0) and passing through an array of squares, a "1" denotes an intersection with a vertical line, while a "0" denotes an intersection with a horizontal line.

Through any A085368(n-1) number of terms in the cutting sequence, A007677(n-1) of those terms are zeros and A007676(n-1) are ones. Check: A085368(4) = 26, that sequence being 3, 4, 11, 15, 26, ... (sum of numerators and denominators of convergents to e). Then, through n = 26, a(n) is 1 1 0 1 1 1 0 1 1 1 0 1 1 0 1 1 1 0 1 1 1 0 1 1 1 0, with 7 zeros and 19 ones, 19/7 = [2; 1, 2, 1, 1] being the 5th convergent to e and 26 being the sum of numerator and denominator (19 + 7).

Positions of zeros and ones are given, respectively, by the Beatty sequences A000572 and A006594.

References

Manfred R. Schroeder, "Fractals, Chaos, Power Laws", Freeman, 1996, p. 56.

Links

Paolo Xausa, Table of n, a(n) for n = 1..10000

Maple

A000572 := proc(n)
    floor(n*(exp(1)+1)) ;
end proc:
isA000572 := proc(n)
    for i from 1 do
        if A000572(i) = n then
            return true ;
        elif A000572(i) > n then
            return false ;
        end if;
    end do:
end proc:
A085369 := proc(n)
    if isA000572(n) then
        0 ;
    else
        1;
    end if;
end proc:
seq(A085369(n), n=1..120) ; # R. J. Mathar, Sep 12 2025

Mathematica

Normal[SparseArray[Map[{#} &, Floor[Range[100]*(1 + 1/E)]] -> 1]] (* Paolo Xausa, Sep 17 2025 *)

Crossrefs

Cf. A085368, A007677, A007676, A006594, A000572.

Sequence in context: A285358 A230603 A229343 * A188082 A046980 A152822

Adjacent sequences: A085366 A085367 A085368 * A085370 A085371 A085372

Keyword

nonn

Author

Gary W. Adamson, Jun 26 2003

Extensions

Edited by Paolo Xausa, Sep 17 2025

Status

approved