OFFSET
0,3
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
These conjectures are still open, although I conjecture that all four are true. This sequence could be called the Heraclitus transform of the squares. The OEIS now contains many other examples of Heraclitus transforms. N. J. A. Sloane, Feb 16 2026
LINKS
N. J. A. Sloane, Table of n, a(n) for n = 0..20000 [First 10000 terms from Rémy Sigrist]
N. J. A. Sloane, Table of n, a(n) for n = 0..250000
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, 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.
EXAMPLE
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
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 *)
PROG
(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)
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
(JavaScript)
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
CROSSREFS
This sequence is a variant of A277616 allowing negative values.
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
KEYWORD
AUTHOR
Rémy Sigrist, Oct 16 2024
STATUS
approved
