A377091 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; 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

0,3

Comments

This sequence is a variant of A277616 allowing negative values.

Will every integer appear in the sequence?

Links

N. J. A. Sloane, Table of n, a(n) for n = 0..20000 [First 10000 terms from Rémy Sigrist]

Rémy Sigrist, Scatterplot of the first 10000 terms of the partial sums

Rémy Sigrist, PARI program

Example

The first 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

Prog

(PARI) \\ See Links section.
(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

Crossrefs

Cf. A277616, A377090, A377092.

Sequence in context: A286548 A256134 A091971 * A341096 A065185 A289412

Adjacent sequences: A377088 A377089 A377090 * A377092 A377093 A377094

Keyword

sign,look,changed

Author

Rémy Sigrist, Oct 16 2024

Status

approved