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

Author

Nicolay Avilov, Nov 09 2022

Keywords

• nonn

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).

Examples

			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).
		

Links

• Nicolay Avilov, Explanatory drawing
• Nicolay Avilov, Multiplication table for sequence

Crossrefs

Cf. A001481, A071253.

Programs

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