A295515 The Euclid tree, read across levels.

0, 1, 1, 1, 1, 2, 2, 1, 1, 3, 3, 2, 2, 3, 3, 1, 1, 4, 4, 3, 3, 5, 5, 2, 2, 5, 5, 3, 3, 4, 4, 1, 1, 5, 5, 4, 4, 7, 7, 3, 3, 8, 8, 5, 5, 7, 7, 2, 2, 7, 7, 5, 5, 8, 8, 3, 3, 7, 7, 4, 4, 5, 5, 1, 1, 6, 6, 5, 5, 9, 9, 4, 4, 11, 11, 7, 7, 10, 10, 3, 3, 11, 11, 8, 8

Offset: 1

Author

Peter Luschny, Nov 25 2017

Keywords

• nonn

Comments

Set N(x) = 1 + floor(x) - frac(x) and let '"' denote the ditto operator, referring to the previously computed expression. Assume the first expression is '0'. Then [0, repeat(N("))] will generate the natural numbers 0, 1, 2, 3, ... and [0, repeat(1/N("))] will generate the rational numbers 0/1, 1/1, 1/2, 2/1, 1/3, 3/2, ... Every reduced nonnegative rational number r appears exactly once in this list as a relatively prime pair [n, d] = r = n/d. We list numerator and denominator one after the other in the sequence.

The apt name 'Euclid tree' is taken from the exposition of Malter, Schleicher and Don Zagier. It is sometimes called the Calkin-Wilf tree. The enumeration is based on Stern's diatomic series (which is a subsequence) and computed by a modification of Dijkstra's 'fusc' function.

The tree listed has root 0, the variant with root 1 is more widely used. Seen as sequences the difference between the two trees is trivial: it is enough to leave out the first two terms; but as trees they are markedly different (see the example section).

Examples

			The tree with root 0 starts:
                                      [0/1]
                  [1/1,                                    1/2]
        [2/1,                1/3,                3/2,                2/3]
   [3/1,      1/4,      4/3,      3/5,      5/2,      2/5,      5/3,      3/4]
[4/1, 1/5, 5/4, 4/7, 7/3, 3/8, 8/5, 5/7, 7/2, 2/7, 7/5, 5/8, 8/3, 3/7, 7/4, 4/5]
.
The tree with root 1 starts:
                                      [1/1]
                  [1/2,                                    2/1]
        [1/3,                3/2,                2/3,                3/1]
   [1/4,      4/3,      3/5,      5/2,      2/5,      5/3,      3/4,      4/1]
[1/5, 5/4, 4/7, 7/3, 3/8, 8/5, 5/7, 7/2, 2/7, 7/5, 5/8, 8/3, 3/7, 7/4, 4/5, 5/1]
		

References

• M. Aigner and G. M. Ziegler, Proofs from The Book, Springer-Verlag, Berlin, 3rd ed., 2004.

Links

• Peter Luschny, Table of n, row(n) for n = 1..12
• N. Calkin and H. S. Wilf, Recounting the rationals, Amer. Math. Monthly, 107 (No. 4, 2000), pp. 360-363.
• Edsger Dijkstra, Selected Writings on Computing, Springer, 1982, p. 232. EWD 578: More about the function 'fusc'.
• Peter Luschny, Rational Trees and Binary Partitions.
• A. Malter, D. Schleicher, and D. Zagier, New looks at old number theory, Amer. Math. Monthly, 120(3), 2013, pp. 243-264.
• Moritz A. Stern, Über eine zahlentheoretische Funktion, J. Reine Angew. Math., 55 (1858), 193-220.
• Index entries for sequences related to enumerating the rationals
• Index entries for sequences related to trees

Crossrefs

Cf. A002487, A174981, A294446 (Stern-Brocot tree), A294442 (Kepler's tree), A295511 (Schinzel-Sierpiński tree), A295512 (encoded by semiprimes).

Programs

Maple

# First implementation: use it only if you are not afraid of infinite loops.
a := x -> 1/(1+floor(x)-frac(x)): 0; do a(%) od;
# Second implementation:
lusc := proc(m) local a, b, n; a := 0; b := 1; n := m; while n > 0 do
if n mod 2 = 1 then b := a + b else a := a + b fi; n := iquo(n, 2) od; a end:
R := n -> 3*2^(n-1)-1 .. 2^n: # The range of level n.
EuclidTree_rat := n -> [seq(lusc(k+1)/lusc(k), k=R(n), -1)]:
EuclidTree_num := n -> [seq(lusc(k+1), k=R(n), -1)]:
EuclidTree_den := n -> [seq(lusc(k), k=R(n), -1)]:
EuclidTree_pair := n -> ListTools:-Flatten([seq([lusc(k+1), lusc(k)], k=R(n), -1)]):
seq(print(EuclidTree_pair(n)), n=1..5);

Sage

def A295515(n):
    if n == 1: return 0
    M = [0, 1]
    for b in (n//2 - 1).bits():
        M[b] = M[0] + M[1]
    return M[1]
print([A295515(n) for n in (1..85)])

Formula

Some characteristics in comparison to the tree with root 1, seen as a table with T(n,k) for n >= 1 and 1 <= k <= 2^(n-1). Here Tr(n,k), Tp(n,k), Tq(n,k) denotes the fraction r, the numerator of r and the denominator of r in row n and column k respectively.

With root 0: With root 1:

Root Tr(1,1) 0/1 1/1

Tp(n,1) 0,1,2,3,... 1,1,1,1,...

Tp(n,2^(n-1)) 0,1,2,3,... 1,2,3,4,...

Tq(n,1) 1,1,1,1,... 1,2,3,4,...

Tq(n,2^(n-1)) 1,2,3,4,... 1,1,1,1,...

Sum_k Tp(n,k) 0,2,8,26,... A024023 1,3,9,27,... A000244

Sum_k Tq(n,k) 1,3,9,27,... A000244 1,3,9,27,... A000244

Sum_k 2Tr(n,k) 0,3,9,21,... A068156 2,5,11,23,... A083329

Sum_k Tp(n,k)Tq(n,k) 0,3,17,81,... A052913-1 1,4,18,82,... A052913

----

a(n) = A002487(floor(n/2)). - Georg Fischer, Nov 29 2022