A023123 Signature sequence of e (arrange the numbers i+j*x (i,j >= 1) in increasing order; the sequence of i's is the signature of x).

1, 2, 3, 1, 4, 2, 5, 3, 6, 1, 4, 7, 2, 5, 8, 3, 6, 9, 1, 4, 7, 10, 2, 5, 8, 11, 3, 6, 9, 1, 12, 4, 7, 10, 2, 13, 5, 8, 11, 3, 14, 6, 9, 1, 12, 4, 15, 7, 10, 2, 13, 5, 16, 8, 11, 3, 14, 6, 17, 9, 1, 12, 4, 15, 7, 18, 10, 2, 13, 5, 16, 8, 19, 11, 3, 14, 6, 17, 9, 20, 1, 12, 4, 15, 7, 18, 10, 21, 2

Offset: 1

Author

Clark Kimberling

Keywords

• nonn
• easy
• nice
• eigen

Comments

If one deletes the first occurrence of 1, the first occurrence of 2, the first occurrence of 3, etc., then the sequence is unchanged. - Brady J. Garvin, Sep 11 2024

Any signature sequence A is closely related to the partial sums of the corresponding homogeneous Beatty sequence: Let Q(d) = d + the sum from g=0 to g=d-1 of floor(theta * g) and Qinv(i) = the maximum integer d such that Q(d) <= i. If there is some d for which Q(d) = i, then A_i = 1. Otherwise, A_i = A_{i - Qinv(i)} + 1. - Brady J. Garvin, Sep 13 2024

References

• Clark Kimberling, "Fractal Sequences and Interspersions", Ars Combinatoria, vol. 45 p 157 1997.

Links

• T. D. Noe, Table of n, a(n) for n=1..1000
• Clark Kimberling, Interspersions
• Index entries for sequences related to signature sequences

Crossrefs

Cf. A023124, A022843

Programs

Mathematica

Quiet[Block[{$ContextPath}, Needs["Combinatorica`"]], {General::compat}]
theta = E;
sums = {0};
cached = <||>;
A023123[i_] := Module[{term, path, base},
  While[sums[[-1]] < i,
    term = sums[[-1]] + Floor[theta * (Length[sums] - 1)] + 1;
    AppendTo[sums, term];
    cached[term] = 1
  ];
  path = {i};
  While[Not[KeyExistsQ[cached, path[[-1]]]],
    AppendTo[path, path[[-1]] - Combinatorica`BinarySearch[sums, path[[-1]]] + 3/2];
  ];
  base = cached[path[[-1]]];
  MapIndexed[(cached[#1] = base + Length[path] - First[#2]) &, path];
  cached[i]
];
Print[Table[A023123[i], {i, 1, 100}]]; (* Brady J. Garvin, Sep 13 2024 *)

Python

from bisect import bisect
from sympy import floor, E
theta = E
sums = [0]
cached = {}
def A023123(i):
    while sums[-1] < i:
        term = sums[-1] + floor(theta * (len(sums) - 1)) + 1
        sums.append(term)
        cached[term] = 1
    path = [i]
    while path[-1] not in cached:
        path.append(path[-1] - bisect(sums, path[-1]) + 1)
    base = cached[path[-1]]
    for offset, vertex in enumerate(reversed(path)):
        cached[vertex] = base + offset
    return cached[i]
print([A023123(i) for i in range(1, 1001)])  # Brady J. Garvin, Sep 13 2024

Extensions

The a(47) term was missing. Corrected by T. D. Noe, Aug 12 2008