cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A379062 First differences of A377091, as in A379061, but take the square root of the magnitude of the difference, keeping the sign.

Original entry on oeis.org

1, 1, -2, 1, 2, 1, 1, -3, 1, 3, 1, 1, -4, 1, 1, 1, -2, -1, -1, -1, 5, -2, 1, 1, 1, -5, -1, -1, -1, -1, -1, 6, -2, 1, 1, 1, -6, -1, -1, -1, -1, -1, 7, -2, -1, -1, 2, -1, 2, 1, 1, -2, -7, -1, -1, -1, -1, -1, -1, -1, 8, -1, -1, -1, 2, 1, 1, 1, 1, 1, 1, 1, 1, -9, 2, 1, 1, 1, -2, -1, -1, 9, 1, 1, 1, 1, 1, 1, 1, 1, -10, 3, -1, -1, -1, -1, -1, -1, -1, -1, 10, 1, 1, 1, 1, 1, 1
Offset: 0

Views

Author

N. J. A. Sloane, Dec 27 2024

Keywords

Comments

By definition, the first differences of A377091 are positive or negative squares.

Links

Crossrefs

Cf. A377091, A379061.

Programs

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.

Original entry on oeis.org

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

Views

Author

Rémy Sigrist, Oct 16 2024

Keywords

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

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

A379066 Indices in A377091 where there is a record upward jump from a negative term to a positive term.

Original entry on oeis.org

0, 4, 9, 20, 31, 42, 60, 81, 100, 121, 147, 183, 210, 241, 272, 307, 342, 400, 441, 484, 529, 576, 651, 703, 757, 813, 871, 931, 1023, 1059, 1125, 1190, 1296, 1369, 1408, 1520, 1598, 1681, 1764, 1849, 1936, 2070, 2209, 2302, 2399, 2500, 2601, 2707, 2862, 2972, 3081, 3192, 3364, 3481, 3600, 3721, 3906, 4033, 4160, 4291, 4422, 4560, 4761, 4900, 5041, 5256, 5403, 5550
Offset: 1

Views

Author

N. J. A. Sloane, Dec 28 2024

Keywords

Comments

These are the indices of record (positive) high points in A379061.
The k-th high point corresponds to a jump in A379061 of height approximately k^2. These heights (whose square roots are 1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, ...) do not warrant an OEIS entry of their own.

Examples

			The third term, 9, corresponds to the jump at the 9th term in A377091 from A377091(9) = -3 to A377091(10) = 6. (The fact that this is a jump of 9 = 3^2 is just a coincidence.)
The fourth term, 20, corresponds to the jump of height 25 = 5^2 at the 20th term in A377091 from A377091(20) = -12 to A377091(21) = 13.

Links

Crossrefs

Cf. A377091, A379061, A379062.
See also A380837 and A380838.
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.