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

1,2

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;
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 *)

Prog

(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

Crossrefs

Cf. A023124, A022843

Sequence in context: A179548 A023119 A118276 * A023131 A375559 A356625

Adjacent sequences: A023120 A023121 A023122 * A023124 A023125 A023126

Keyword

nonn,easy,nice,eigen

Author

Clark Kimberling

Extensions

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

Status

approved