A023124 Signature sequence of 1/e (arrange the numbers i+j*x (i,j >= 1) in increasing order; the sequence of i's is the signature of x).

1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 2, 1, 3, 2, 1, 3, 2, 1, 4, 3, 2, 1, 4, 3, 2, 1, 4, 3, 2, 5, 1, 4, 3, 2, 5, 1, 4, 3, 2, 5, 1, 4, 3, 6, 2, 5, 1, 4, 3, 6, 2, 5, 1, 4, 3, 6, 2, 5, 1, 4, 7, 3, 6, 2, 5, 1, 4, 7, 3, 6, 2, 5, 1, 4, 7, 3, 6, 2, 5, 1, 8, 4, 7, 3, 6, 2, 5, 1, 8, 4, 7, 3, 6, 2, 5, 1, 8, 4, 7, 3

Offset

1,4

Comments

Arrange the numbers i+j*e (i,j >= 1) in increasing order; this sequence is the sequence of j's. - Michel Marcus, Dec 18 2021

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

J.-P. Delahaye, Des suites fractales d’entiers, Pour la Science, No. 531 January 2022. Sequence h) p. 82.

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

Mathematica

Quiet[Block[{$ContextPath}, Needs["Combinatorica`"]], {General::compat}]
theta = 1 / E;
sums = {0};
cached = <||>;
A023124[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[A023124[i], {i, 1, 100}]]; (* Brady J. Garvin, Sep 13 2024 *)

Prog

(Python)
from bisect import bisect
from sympy import floor, E
theta = 1 / E
sums = [0]
cached = {}
def A023124(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([A023124(i) for i in range(1, 1001)])  # Brady J. Garvin, Sep 13 2024

Crossrefs

Cf. A001113, A023123, A032634.

Sequence in context: A213649 A242397 A023132 * A023120 A167970 A126433

Adjacent sequences: A023121 A023122 A023123 * A023125 A023126 A023127

Keyword

nonn,easy,nice,eigen

Author

Clark Kimberling

Status

approved