A379057 -id:A379057 - cp's OEIS frontend

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

Rémy Sigrist, Oct 16 2024

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

			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

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.

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

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

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

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.
```

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

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

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

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

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A379058 First differences of A379057.

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A380313 Partial sums of A379057.

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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.

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