A277616 -id:A277616 - 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
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\\ 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

A277618 Lexicographically earliest nonnegative sequence such that |a(n+1)-a(n)| is a prime number, and no number occurs twice; a(0) = 0.

Original entry on oeis.org

0, 2, 4, 1, 3, 5, 7, 9, 6, 8, 10, 12, 14, 11, 13, 15, 17, 19, 16, 18, 20, 22, 24, 21, 23, 25, 27, 29, 26, 28, 30, 32, 34, 31, 33, 35, 37, 39, 36, 38, 40, 42, 44, 41, 43, 45, 47, 49, 46, 48, 50, 52, 54, 51, 53, 55, 57, 59, 56, 58, 60, 62, 64, 61, 63, 65, 67, 69, 66, 68, 70, 72, 74, 71, 73, 75, 77, 79, 76, 78, 80, 82, 84, 81, 83, 85, 87, 89, 86, 88, 90

Offset: 0

Eric Angelini and M. F. Hasler, Oct 23 2016

A variant of A277616, which is defined in a similar way with squares > 1 instead of primes.

The steps a(n+1)-a(n) are either +2 or -3, after 10 terms we get a(n+10), and the first 10 terms are all numbers between 0 and 9: This sequence is obviously a permutation of the nonnegative integers, with a(n) = a(n-5) + 5 for all n > 5. The strictly positive variant (starting with a(1) = 1) is given by A065186.

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1).

Cf. A277616, A277617, A065186.

Table[n - 2 + Mod[n - 3, 5], {n, 0, 500}] (* Fred Patrick Doty, Aug 03 2020 *)

{u=[a=0];(chk(n)=(!#u||(n>u[1]&&!setsearch(u,n)))&&(u=setunion(u,[n]))&&!while(#u>1&&u[2]==u[1]+1,u=u[^1]));for(n=1,99,print1(a","); for(k=-primepi(a+!a-1),9e9, k||next; chk(a+sign(k)*prime(abs(k)))||next; a+=sign(k)*prime(abs(k));break))}

A277618(n,i=[0,2,4,1,3])=i[n%#i+1]+n\#i*#i

a(n) = a(n-5) + 5 for all n > 5.
a(n) = A065186(n+1) - 1.
G.f.: x*(2*x^4 + 2*x^3 - 3*x^2 + 2*x + 2)/(x^6 - x^5 - x + 1). - Chai Wah Wu, Mar 30 2023

Offset corrected by R. J. Mathar, Jun 19 2021

A277617 Lexicographically earliest positive sequence such that a(n+1)-a(n) is a square > 1 and no number occurs twice; a(1) = 1.

Original entry on oeis.org

1, 5, 9, 13, 4, 8, 12, 3, 7, 11, 2, 6, 10, 14, 18, 22, 26, 17, 21, 25, 16, 20, 24, 15, 19, 23, 27, 31, 35, 39, 30, 34, 38, 29, 33, 37, 28, 32, 36, 40, 44, 48, 52, 43, 47, 51, 42, 46, 50, 41, 45, 49, 53, 57, 61, 65, 56, 60, 64, 55, 59, 63, 54, 58, 62, 66, 70, 74, 78, 69, 73, 77, 68, 72, 76, 67, 71, 75, 79, 83, 87, 91, 82, 86, 90, 81, 85, 89, 80, 84, 88

Offset: 1

Eric Angelini and M. F. Hasler, Oct 23 2016

A variant of A277616, which is defined in the same way but starts with a(0) = 0.

Another variant is A277618, which is defined in a similar way, but with primes instead of squares. (The strictly positive variant is A065186.)

Paolo Xausa, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,0,0,0,1,-1).

Cf. A277616, A277618, A065186.

LinearRecurrence[{1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1}, {1, 5, 9, 13, 4, 8, 12, 3, 7, 11, 2, 6, 10, 14}, 100] (* Paolo Xausa, Jan 16 2025 *)

{u=[a=1];(chk(n)=(!#u||(n>u[1]&&!setsearch(u,n)))&&(u=setunion(u,[n]))&&!while(#u>1&&u[2]==u[1]+1,u=u[^1]));for(n=1,99,print1(a",");for(k=-sqrtint(a+!a-1),9e9,k^2>1||next;chk(a+k*abs(k))||next;a+=k*abs(k);break))}

From Chai Wah Wu, Mar 30 2023: (Start)
a(n) = a(n-1) + a(n-13) - a(n-14) for n > 14.
G.f.: x*(3*x^13 + 4*x^12 + 4*x^11 - 9*x^10 + 4*x^9 + 4*x^8 - 9*x^7 + 4*x^6 + 4*x^5 - 9*x^4 + 4*x^3 + 4*x^2 + 4*x + 1)/(x^14 - x^13 - x + 1). (End)

A379057 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 > 1; in case of a tie, preference is given to the positive value.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Dec 25 2024

sign

Inspired by A377091 and A277616.

Michael S. Branicky, Table of n, a(n) for n = 0..10000

Cf. A277616, A377091, A379058 (first differences), A380313 (partial sums).

A379057list[nmax_] := Module[{s, a, u = 1}, s[_] := False; s[0] = True; NestList[(While[s[u] && s[-u], u++]; a = u; While[s[a] || Abs[# - a] <= 1 || !IntegerQ[Sqrt[Abs[# - a]]], a = Boole[a < 0] - a]; s[a] = True; a) &, 0, nmax]];
A379057list[100] (* Paolo Xausa, Mar 21 2025 *)

from math import isqrt
from itertools import count, islice
def cond(n): return n > 1 and 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

More terms from Michael S. Branicky, Dec 25 2024

A379675 a(0) = 0, and for any n > 0, a(n) is the least integer (in absolute value) not yet in the sequence such that the absolute difference of a(n-1) and a(n) is a square and a(n) is positive iff n is odd.

Original entry on oeis.org

0, 1, -3, 6, -10, 15, -1, 3, -6, 10, -15, 21, -4, 5, -11, 14, -2, 2, -7, 9, -16, 20, -5, 4, -12, 13, -23, 26, -38, 11, -14, 22, -27, 37, -44, 56, -8, 8, -17, 19, -30, 34, -47, 17, -19, 30, -34, 47, -53, 28, -21, 43, -57, 7, -9, 16, -20, 29, -35, 46, -18, 18

Offset: 0

Rémy Sigrist, Dec 29 2024

sign

This sequence is a variant of A377091 where, beyond the initial term a(0) = 0, the sign of the terms alternates.

Will every integer appear in the sequence?

			The first terms are:
  n   a(n)  |a(n)-a(n-1)|
  --  ----  -------------
   0     0  N/A
   1     1  1^2
   2    -3  2^2
   3     6  3^2
   4   -10  4^2
   5    15  5^2
   6    -1  4^2
   7     3  2^2
   8    -6  3^2
   9    10  4^2
  10   -15  5^2
  11    21  6^2
  12    -4  5^2
  13     5  3^2
  14   -11  4^2

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Rémy Sigrist, Colored scatterplot of the sequence for n = 0..10000 (where the color is function of Sum_{k = 0..n-1} sign(a(k+1)-a(k)) * sqrt(abs(a(k+1)-a(k))))
Rémy Sigrist, PARI program

Cf. A277616, A377091.

PARI
```
\\ See Links section.
```

A344538 Lexicographically earliest sequence such that |a(n+1)-a(n)| is a cube > 1 and no number occurs twice; a(0) = 0.

Original entry on oeis.org

0, 8, 16, 24, 32, 5, 13, 21, 29, 2, 10, 18, 26, 34, 7, 15, 23, 31, 4, 12, 20, 28, 1, 9, 17, 25, 33, 6, 14, 22, 30, 3, 11, 19, 27, 35, 43, 51, 59, 67, 40, 48, 56, 64, 37, 45, 53, 61, 69, 42, 50, 58, 66, 39, 47, 55, 63, 36, 44, 52, 60, 68, 41, 49, 57, 65, 38

Offset: 0

Mike Koss, May 22 2021

The difference between successive terms, a(n+1)-a(n), is either +8 or -27. This sequence is a permutation of the nonnegative integers (with no "gaps").

Mike Koss, Derivation and proof.
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,-1,2,-1,0,0,0,0,-1,2,-1,0,0,0,0,-1,2,-1,0,0,0,0,-1,2,-1).
Index entries for sequences that are permutations of the nonnegative integers

Cf. A277616.

a[0]=0;a[n_]:=a[n]=(k=1;While[!IntegerQ[s=Abs[k-a[n-1]]^(1/3)]||s==1||MemberQ[Array[a,n-1],k],k++];k);Array[a,100,0] (* Giorgos Kalogeropoulos, May 27 2021 *)

a(n+35) = a(n) + 35 for all n.

a(n) = 2*a(n-1) - a(n-2) - a(n-7) + 2*a(n-8) - a(n-9) - a(n-14) + 2*a(n-15) - a(n-16) - a(n-21) + 2*a(n-22) - a(n-23) - a(n-28) + 2*a(n-29) - a(n-30) for n > 29. - Stefano Spezia, May 23 2021

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.

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A277618 Lexicographically earliest nonnegative sequence such that |a(n+1)-a(n)| is a prime number, and no number occurs twice; a(0) = 0.

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A277617 Lexicographically earliest positive sequence such that a(n+1)-a(n) is a square > 1 and no number occurs twice; a(1) = 1.

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A379057 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 > 1; in case of a tie, preference is given to the positive value.

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A379675 a(0) = 0, and for any n > 0, a(n) is the least integer (in absolute value) not yet in the sequence such that the absolute difference of a(n-1) and a(n) is a square and a(n) is positive iff n is odd.

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A344538 Lexicographically earliest sequence such that |a(n+1)-a(n)| is a cube > 1 and no number occurs twice; a(0) = 0.

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