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A238456
Triangular numbers t such that t+x+y is a square, where x and y are the two squares nearest to t.
2
0, 2211, 5151, 1107816, 20959575, 4237107540, 1564279847151, 61066162885575, 2533192954461975, 2774988107938203, 90728963274006291, 18765679728507154152720
OFFSET
1,2
COMMENTS
For triangular numbers t such that t*x*y is a square, see A001110 (t is both triangular and square).
a(13) > 5*10^22. - Giovanni Resta, Mar 02 2014
EXAMPLE
The two squares nearest to triangular(101)=5151 are 71^2 and 72^2. Because 5151 + 71^2 + 72^2 = 15376 is a perfect square, 5151 is in the sequence.
MATHEMATICA
sqQ[n_]:=Module[{c=Floor[Sqrt[n]]-1, x}, x=Total[Take[SortBy[ Range[ c, c+3]^2, Abs[#-n]&], 2]]; IntegerQ[Sqrt[n+x]]]; Select[ Accumulate[ Range[ 0, 5000000]], sqQ] (* This will generate the first 7 terms of the sequence. To generate more, increase the second constant within the Range function, but computations will take a long time. *) (* Harvey P. Dale, May 12 2014 *)
PROG
(Python)
def isqrt(a):
sr = 1 << (int.bit_length(int(a)) >> 1)
while a < sr*sr: sr>>=1
b = sr>>1
while b:
s = sr + b
if a >= s*s: sr = s
b>>=1
return sr
t = i = 0
while 1:
t += i
i += 1
s = isqrt(t)
if s*s==t: s-=1
txy = t + 2*s*(s+1) + 1 # t + s^2 + (s+1)^2
r = isqrt(txy)
if r*r==txy: print(str(t), end=', ')
CROSSREFS
KEYWORD
nonn,hard,more
AUTHOR
Alex Ratushnyak, Feb 26 2014
EXTENSIONS
a(12) from Giovanni Resta, Mar 02 2014
STATUS
approved