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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.

%I A358317 #55 Nov 24 2022 18:27:27
%S A358317 0,2,4,10,16,18,20,26,36,50,64,68,80,82,90,98,100,122,144,148,162,170,
%T A358317 180,196,226,234,242,250,256,260,272,290,320,324,338,362,400,404,442,
%U A358317 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
%N 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.
%C A358317 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).
%C A358317 Numbers of the form m^2*(k^2 + 1) for integers m and k of the same parity.
%C A358317 Chords starting at the origin (z=0, or m=k) are terms A071253(x).
%H A358317 Nicolay Avilov, <a href="/A358317/a358317.jpg">Explanatory drawing</a>
%H A358317 Nicolay Avilov, <a href="/A358317/a358317_1.jpg">Multiplication table for sequence</a>
%e A358317 0 is a term since it is the square of the chord length from (0,0) to (0,0).
%e A358317 10 = 1^2 + 3^2 is a term since it is the square of the chord length from (1,1) to (2,4).
%o A358317 (Python)
%o A358317 # Program from Oleg Sorokin
%o A358317 from math import isqrt
%o A358317 limit = 2000
%o A358317 s = set()
%o A358317 end = isqrt(limit)
%o A358317 for m in range(0, end+1):
%o A358317     for k in range(m%2, end+1, 2):
%o A358317         c = m**2*(k**2+1)
%o A358317         if c > limit:
%o A358317             break
%o A358317         s.add(c)
%o A358317 print(sorted(s))
%o A358317 (Python)
%o A358317 from itertools import count, islice
%o A358317 from sympy import divisors, integer_nthroot
%o A358317 def A358317_gen(startvalue=0): # generator of terms >= startvalue
%o A358317     for n in count(max(startvalue,0)):
%o A358317         if n == 0:
%o A358317             yield 0
%o A358317         else:
%o A358317             for d in divisors(n,generator=True):
%o A358317                 a, b = integer_nthroot(d,2)
%o A358317                 if b:
%o A358317                     c, e = integer_nthroot(n//d-1,2)
%o A358317                     if e and not (c^a)&1:
%o A358317                         yield n
%o A358317                         break
%o A358317 A358317_list = list(islice(A358317_gen(),30)) # _Chai Wah Wu_, Nov 24 2022
%Y A358317 Cf. A001481, A071253.
%K A358317 nonn
%O A358317 1,2
%A A358317 _Nicolay Avilov_, Nov 09 2022