OFFSET
1,2
COMMENTS
If k is a square then y=k.
The sequence of terms that are perfect squares begins: 0, 4, 576, 108900, 21086464, 4090114116, 793453377600.
In other words, numbers x such that x + 2y(y+1) = z^2 has a solution with x in the interval [y^2+1, (y+1)^2], see Sage program. - Ralf Stephan, Mar 09 2014
The nonzero terms which are perfect squares are exactly the squares of A081065. - Ivan Neretin, Jun 25 2015
LINKS
Harvey P. Dale, Table of n, a(n) for n = 1..500
EXAMPLE
The two squares nearest to 4 are 1 and 4. Because 4+1+4=9 is a square, 4 is in the sequence.
The two squares nearest to 11 are 9 and 16. Because 11+9+16=36 is a square, 11 is in the sequence.
MATHEMATICA
kxyQ[n_]:=Module[{c=Floor[Sqrt[n]]}, IntegerQ[Sqrt[n+Total[Nearest[Range[c-2, c+2]^2, n, 2]]]]]; Join[{0}, Select[Range[3, 7500], kxyQ]] (* Harvey P. Dale, Apr 24 2022 *)
PROG
(Python) # use version >= 3.8
from math import isqrt
for k in range(7777):
s = isqrt(k)
if s*s==k: s-=1
kxy = k + 2*s*(s+1) + 1 # k + s^2 + (s+1)^2
r = isqrt(kxy)
if r*r==kxy: print(str(k), end=', ')
(Sage)
def gen_a():
n = 1
while True:
for t in range(n*n + 1, n*n + 2*n + 2):
if is_square(t + 2*(n*n + n) + 1):
yield t
n += 1 # Ralf Stephan, Mar 09 2014
CROSSREFS
KEYWORD
nonn
AUTHOR
Alex Ratushnyak, Feb 27 2014
STATUS
approved