A383442 a(0) = 0; thereafter a(n) is the least integer (in absolute value) not yet in the sequence such that the absolute difference between a(n-1) and a(n) is a triangular number; in case of a tie, preference is given to the positive value.

0, 1, 2, -1, -2, -3, 3, 4, 5, -5, -4, 6, 7, 8, -7, -6, 9, 10, 11, -10, -9, -8, -11, -12, -13, -14, 14, 13, 12, 15, 16, 17, 18, -18, -15, -16, -17, 19, 20, 21, 22, 23, -22, -19, -20, -21, 24, 25, 26, 27, 28, -27, -24, -23, -26, -25, -28, -29, -30, -31, -32, -33, 33, 30, 29, 32, 31, 34, 35, 36, 37, 38, 39, -39, -36, -35, -34, -37, -38, 40

Offset: 0

Author

N. J. A. Sloane, May 03 2025

Keywords

• sign

Comments

Heraclitus (circa 500 BCE) observed that no man can step in the same river twice.

The Heraclitus transform H(S) of a sequence S is formed by starting at 0, and moving s steps to the left or right, where s is any element of S, never visiting any number twice, and moving as close to 0 as possible. In case of a tie, move to the positive term.

The present sequence is the Heraclitus transform of the triangular numbers A000217. For the squares, see A377091. Conjecture: both H(A000217) and H(A000290) contain every (positive or negative) integer. In fact it appears that this property holds whenever S is a monotonically strictly increasing sequence starting with 1. It does not hold for H(A000012), which is A001477.

Links

• Michael S. Branicky, Table of n, a(n) for n = 0..20000

Crossrefs

Cf. A000217, A000290, A001477, A377091, A383443, A383688.

Programs

Mathematica

A383442list[nmax_] := Module[{s, a, u = 1}, s[_] := False; s[0] = True; NestList[(While[s[u] && s[-u], u++]; a = u; While[s[a] || !IntegerQ[Sqrt[8*Abs[# - a]+1]], a = Boole[a < 0] - a]; s[a] = True; a) &, 0, nmax]];
A383442list[100] (* Paolo Xausa, May 05 2025 *)

Python

from math import isqrt
from itertools import count, islice
def cond(n): return isqrt(m:=8*n+1)**2 == m
def agen(): # generator of terms
    an, aset, m = 0, {0}, 1
    for n in count(0):
        yield an
        an = next(s for k in count(m) for s in [k, -k] if s not in aset and cond(abs(an-s)))
        aset.add(an)
        while m in aset and -m in aset: m += 1
print(list(islice(agen(), 80))) # Michael S. Branicky, May 03 2025