A238456 Triangular numbers t such that t+x+y is a square, where x and y are the two squares nearest to t.

0, 2211, 5151, 1107816, 20959575, 4237107540, 1564279847151, 61066162885575, 2533192954461975, 2774988107938203, 90728963274006291, 18765679728507154152720

Offset: 1

Author

Alex Ratushnyak, Feb 26 2014

Keywords

• nonn
• hard
• more

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

Examples

			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.
		

Crossrefs

Cf. A000217, A000290, A001110, A238489.

Programs

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 *)

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=',')

Extensions

a(12) from Giovanni Resta, Mar 02 2014