A023125 Signature sequence of e^2 (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, 4, 5, 6, 7, 8, 1, 9, 2, 10, 3, 11, 4, 12, 5, 13, 6, 14, 7, 15, 8, 1, 16, 9, 2, 17, 10, 3, 18, 11, 4, 19, 12, 5, 20, 13, 6, 21, 14, 7, 22, 15, 8, 23, 1, 16, 9, 24, 2, 17, 10, 25, 3, 18, 11, 26, 4, 19, 12, 27, 5, 20, 13, 28, 6, 21, 14, 29, 7, 22, 15, 30, 8, 23, 1, 16

Offset

1,2

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

Mathematica

Quiet[Block[{$ContextPath}, Needs["Combinatorica`"]], {General::compat}]
theta = E * E;
sums = {0};
cached = <||>;
A023125[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[A023125[i], {i, 1, 100}]]; (* Brady J. Garvin, Sep 13 2024 *)

Prog

(Python)
from bisect import bisect
from sympy import floor, E
theta = E * E
sums = [0]
cached = {}
def A023125(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([A023125(i) for i in range(1, 1001)])  # Brady J. Garvin, Sep 13 2024

Crossrefs

Cf. A023126

Sequence in context: A033929 A025482 A375481 * A319657 A255594 A030108

Adjacent sequences: A023122 A023123 A023124 * A023126 A023127 A023128

Keyword

nonn,easy,eigen

Author

Clark Kimberling

Status

approved