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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.

%I A377091 #169 Apr 13 2025 00:51:34
%S A377091 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,
%T A377091 12,-13,-14,-15,-16,-17,-18,18,14,15,16,17,-19,-20,-21,-22,-23,-24,25,
%U A377091 21,20,19,23,22,26,27,28,24,-25,-26,-27,-28,-29,-30,-31,-32,32
%N 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.
%C A377091 Conjecture 1: Every integer (positive or negative) appears in this sequence.
%C A377091 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]
%C A377091 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
%C A377091 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
%H A377091 N. J. A. Sloane, <a href="/A377091/b377091.txt">Table of n, a(n) for n = 0..20000</a> [First 10000 terms from Rémy Sigrist]
%H A377091 M. F. Hasler, <a href="/A377091/a377091_2.html">Interactive spiral graph representation of the first n terms</a>, Jan 18 2025.
%H A377091 Kerry Mitchell, <a href="/A377091/a377091_2.jpg">Spiral graph representation of first 31 terms</a>
%H A377091 Kerry Mitchell, <a href="/A377091/a377091_3.jpg">Colored spiral graph representation of first 100 terms</a>
%H A377091 Chris Scussel, <a href="/A377091/a377091.pdf">Spiral graph representation of 5000 terms</a>
%H A377091 Rémy Sigrist, <a href="/A377091/a377091.png">Scatterplot of the first 10000 terms of the partial sums</a>
%H A377091 Rémy Sigrist, <a href="/A377091/a377091_1.png">Christmas card based on graph of the partial sums</a> [Rotate graph by 90 degs, add color and decorations]
%H A377091 Rémy Sigrist, <a href="/A377091/a377091.gp.txt">PARI program</a>
%H A377091 N. J. A. Sloane, <a href="/A377091/a377091.txt">Table of n, a(n) for n = 0..250000</a>
%H A377091 N. J. A. Sloane, <a href="/A377091/a377091_1.jpg">Spiral graph representation of first 28 terms</a> [Hand-drawn]
%H A377091 Paolo Xausa, <a href="/A377091/a377091_2.png">Scatterplot of ||a(n)| - floor(n/2)| for 0 <= n <= 200000.</a> The orange line is sqrt(n)/2.
%e A377091 The initial terms are:
%e A377091   n   a(n)  |a(n)-a(n-1)|
%e A377091   --  ----  -------------
%e A377091    0     0  N/A
%e A377091    1     1  1^2
%e A377091    2     2  1^2
%e A377091    3    -2  2^2
%e A377091    4    -1  1^2
%e A377091    5     3  2^2
%e A377091    6     4  1^2
%e A377091    7     5  1^2
%e A377091    8    -4  3^2
%e A377091    9    -3  1^2
%e A377091   10     6  3^2
%e A377091   11     7  1^2
%e A377091   12     8  1^2
%e A377091   13    -8  4^2
%e A377091   14    -7  1^2
%p A377091 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:
%p A377091 a_list := length -> h(length, [0], 1): a_list(62);  # _Peter Luschny_, Jan 20 2025
%t A377091 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]];
%t A377091 A377091list[100] (* _Paolo Xausa_, Mar 18 2025 *)
%o A377091 (PARI) \\ See Links section.
%o A377091 (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
%o A377091 (Python)
%o A377091 from math import isqrt
%o A377091 from itertools import count, islice
%o A377091 def cond(n): return isqrt(n)**2 == n
%o A377091 def agen(): # generator of terms
%o A377091     an, aset, m = 0, {0}, 1
%o A377091     for n in count(0):
%o A377091         yield an
%o A377091         an = next(s for k in count(m) for s in [k, -k] if s not in aset and cond(abs(an-s)))
%o A377091         aset.add(an)
%o A377091         while m in aset and -m in aset: m += 1
%o A377091 print(list(islice(agen(), 62))) # _Michael S. Branicky_, Dec 25 2024
%o A377091 (Python)
%o A377091 from math import sqrt
%o A377091 def a_list(b: int, a: list[int] = [0], i: int = 1) -> list[int]:
%o A377091     if sqrt(abs(a[-1] - i)).is_integer() and not (i in a):
%o A377091         a += [i]
%o A377091         if b < len(a):
%o A377091             return a
%o A377091         else:
%o A377091             return a_list(b, a)
%o A377091     else:
%o A377091         return a_list(b, a, int(i < 0) - i)
%o A377091 print(a_list(40))  # _Peter Luschny_, Jan 20 2025
%o A377091 (Python)
%o A377091 class A377091: # A377091(n) gives a(n)
%o A377091     terms = [0]; N = 1 # next candidate
%o A377091     def __new__(A, n): A.extend(A, n-len(A.terms)+1); return A.terms[n]
%o A377091     def extend(A, n): any((k:=A.N) in A.terms and setattr(A, 'N', k:=(k<0)-k) or
%o A377091         A.terms.append(next(k for _ in range(9**9) if (abs(A.terms[-1]-k)**.5)
%o A377091        .is_integer() and k not in A.terms or not(k:=(k<0)-k))) for _ in range(n))
%o A377091 # _M. F. Hasler_, Feb 08 2025
%o A377091 (JavaScript)
%o A377091 A377091 = [0]; A377091.least_unused = 1;
%o A377091 function a(n){
%o A377091   for(let i = A377091.length-1; i < n; ++i) {
%o A377091     let k = A377091.least_unused;
%o A377091     while(!Number.isInteger(Math.sqrt(Math.abs(A377091[i] - k)))
%o A377091           || A377091.indexOf(k) > 0) k = (k<0)-k;
%o A377091     A377091.push(k);
%o A377091     if (k == A377091.least_unused) {
%o A377091       do k = (k<0)-k; while ( A377091.indexOf( k ) > 0 );
%o A377091       A377091.least_unused = k;
%o A377091   } };
%o A377091   return A377091[n];
%o A377091 } // _M. F. Hasler_, Jan 26 2025
%Y A377091 This sequence is a variant of A277616 allowing negative values.
%Y A377091 Cf. A277617, A277618, A377090, A377092.
%Y A377091 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.
%Y A377091 First differences are A379061 (certainly the most relevant derived sequence). - _M. F. Hasler_, Feb 08 2025
%Y A377091 "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
%K A377091 sign,look,nice
%O A377091 0,3
%A A377091 _Rémy Sigrist_, Oct 16 2024