A358317 Ordered squares of the chord lengths of the parabola y=x^2, where the chord ends are all possible points of the parabola with integer coordinates.

0, 2, 4, 10, 16, 18, 20, 26, 36, 50, 64, 68, 80, 82, 90, 98, 100, 122, 144, 148, 162, 170, 180, 196, 226, 234, 242, 250, 256, 260, 272, 290, 320, 324, 338, 362, 400, 404, 442, 450, 484, 490, 500, 530, 576, 578, 580, 592, 612, 626, 650, 676, 720, 722, 730, 738, 784, 788, 810, 842, 882, 900, 962, 980

Offset

1,2

Comments

Numbers of the form (x^2 - z^2)^2 + (x-z)^2 for integers x and z, so that terms are sums of 2 squares (subset of A001481).

Numbers of the form m^2*(k^2 + 1) for integers m and k of the same parity.

Chords starting at the origin (z=0, or m=k) are terms A071253(x).

Links

Table of n, a(n) for n=1..64.

Nicolay Avilov, Explanatory drawing

Nicolay Avilov, Multiplication table for sequence

Example

0 is a term since it is the square of the chord length from (0,0) to (0,0).
10 = 1^2 + 3^2 is a term since it is the square of the chord length from (1,1) to (2,4).

Prog

(Python)
# Program from Oleg Sorokin
from math import isqrt
limit = 2000
s = set()
end = isqrt(limit)
for m in range(0, end+1):
    for k in range(m%2, end+1, 2):
        c = m**2*(k**2+1)
        if c > limit:
            break
        s.add(c)
print(sorted(s))
(Python)
from itertools import count, islice
from sympy import divisors, integer_nthroot
def A358317_gen(startvalue=0): # generator of terms >= startvalue
    for n in count(max(startvalue, 0)):
        if n == 0:
            yield 0
        else:
            for d in divisors(n, generator=True):
                a, b = integer_nthroot(d, 2)
                if b:
                    c, e = integer_nthroot(n//d-1, 2)
                    if e and not (c^a)&1:
                        yield n
                        break
A358317_list = list(islice(A358317_gen(), 30)) # Chai Wah Wu, Nov 24 2022

Crossrefs

Cf. A001481, A071253.

Sequence in context: A000947 A125245 A371896 * A096689 A039682 A218665

Adjacent sequences: A358314 A358315 A358316 * A358318 A358319 A358320

Keyword

nonn

Author

Nicolay Avilov, Nov 09 2022

Status

approved