A377091 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 square; in case of a tie, preference is given to the positive value.

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

Offset: 0

Author

Rémy Sigrist, Oct 16 2024

Keywords

• sign
• look
• nice

Comments

Conjecture 1: Every integer (positive or negative) appears in this sequence.

Conjecture 2: For n > 16, |a(n)| is within sqrt(n/2) of floor(n/2). See A379071. - N. J. A. Sloane, Dec 29 2024 [Corrected by Paolo Xausa, Jan 21 2025]

Conjecture 3: lim sup ||a(n)| - floor(n/2)|/sqrt(n) = 1/2. (See link.) - N. J. A. Sloane and Paolo Xausa, Feb 03 2025

Conjecture 4: After a(n) has been found, the sequence contains all numbers in the range [0,f(n)], where lim sup f(n) = (n-sqrt(n))/2. There is a corresponding conjecture for the negative terms. See A379067. - N. J. A. Sloane and Paolo Xausa, Feb 13 2025

Examples

			The initial terms are:
  n   a(n)  |a(n)-a(n-1)|
  --  ----  -------------
   0     0  N/A
   1     1  1^2
   2     2  1^2
   3    -2  2^2
   4    -1  1^2
   5     3  2^2
   6     4  1^2
   7     5  1^2
   8    -4  3^2
   9    -3  1^2
  10     6  3^2
  11     7  1^2
  12     8  1^2
  13    -8  4^2
  14    -7  1^2
		

Links

• N. J. A. Sloane, Table of n, a(n) for n = 0..20000 [First 10000 terms from Rémy Sigrist]
• M. F. Hasler, Interactive spiral graph representation of the first n terms, Jan 18 2025.
• Kerry Mitchell, Spiral graph representation of first 31 terms
• Kerry Mitchell, Colored spiral graph representation of first 100 terms
• Chris Scussel, Spiral graph representation of 5000 terms
• Rémy Sigrist, Scatterplot of the first 10000 terms of the partial sums
• Rémy Sigrist, Christmas card based on graph of the partial sums [Rotate graph by 90 degs, add color and decorations]
• Rémy Sigrist, PARI program
• N. J. A. Sloane, Table of n, a(n) for n = 0..250000
• N. J. A. Sloane, Spiral graph representation of first 28 terms [Hand-drawn]
• Paolo Xausa, Scatterplot of ||a(n)| - floor(n/2)| for 0 <= n <= 200000. The orange line is sqrt(n)/2.

Crossrefs

This sequence is a variant of A277616 allowing negative values.

Cf. A277617, A277618, A377090, A377092.

A large number of sequences have been derived from the present sequence in the hope (so far unfulfilled) of finding a formula or recurrence: see A379057-A379078, A379786-A379798, A379802, A379803, A379804, A379880, A380223, A380224, A380225, A382715-A382718.

First differences are A379061 (certainly the most relevant derived sequence). - M. F. Hasler, Feb 08 2025

"Lexicographically earliest" sequences for which there is a proof that every number that could appear does appear: A064413, A098550, A109812, A121216, A347113, etc. - N. J. A. Sloane, Feb 08 2025

Programs

JavaScript

A377091 = [0]; A377091.least_unused = 1;
function a(n){
  for(let i = A377091.length-1; i < n; ++i) {
    let k = A377091.least_unused;
    while(!Number.isInteger(Math.sqrt(Math.abs(A377091[i] - k)))
          || A377091.indexOf(k) > 0) k = (k<0)-k;
    A377091.push(k);
    if (k == A377091.least_unused) {
      do k = (k<0)-k; while ( A377091.indexOf( k ) > 0 );
      A377091.least_unused = k;
  } };
  return A377091[n];
} // M. F. Hasler, Jan 26 2025

Maple

h := proc(b, a, i) option remember; ifelse(issqr(abs(a[-1] - i)) and not is(i in a), ifelse(b < nops(a) + 1, a, h(b, [op(a), i], 1)), h(b, a, ifelse(i < 0, 1 - i, -i))) end:
a_list := length -> h(length, [0], 1): a_list(62);  # Peter Luschny, Jan 20 2025

Mathematica

A377091list[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[Abs[# - a]]], a = Boole[a < 0] - a]; s[a] = True; a) &, 0,nmax]];
A377091list[100] (* Paolo Xausa, Mar 18 2025 *)

PARI

\\ See Links section.

PARI

A377091_upto(n,S=[])={vector(n+1, k, S=setunion(S, [n=if(k>1, k=1; while(setsearch(S,k) || !issquare(abs(n-k)), k=(k<0)-k); k)]); n)} \\ M. F. Hasler, Jan 18 2025

Python

from math import isqrt
from itertools import count, islice
def cond(n): return isqrt(n)**2 == n
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(), 62))) # Michael S. Branicky, Dec 25 2024

Python

from math import sqrt
def a_list(b: int, a: list[int] = [0], i: int = 1) -> list[int]:
    if sqrt(abs(a[-1] - i)).is_integer() and not (i in a):
        a += [i]
        if b < len(a):
            return a
        else:
            return a_list(b, a)
    else:
        return a_list(b, a, int(i < 0) - i)
print(a_list(40))  # Peter Luschny, Jan 20 2025

Python

class A377091: # A377091(n) gives a(n)
    terms = [0]; N = 1 # next candidate
    def _new_(A, n): A.extend(A, n-len(A.terms)+1); return A.terms[n]
    def extend(A, n): any((k:=A.N) in A.terms and setattr(A, 'N', k:=(k<0)-k) or
        A.terms.append(next(k for _ in range(9**9) if (abs(A.terms[-1]-k)**.5)
       .is_integer() and k not in A.terms or not(k:=(k<0)-k))) for _ in range(n))
# M. F. Hasler, Feb 08 2025