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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.

%I A238456 #19 May 05 2021 18:20:07
%S A238456 0,2211,5151,1107816,20959575,4237107540,1564279847151,61066162885575,
%T A238456 2533192954461975,2774988107938203,90728963274006291,
%U A238456 18765679728507154152720
%N A238456 Triangular numbers t such that t+x+y is a square, where x and y are the two squares nearest to t.
%C A238456 For triangular numbers t such that t*x*y is a square, see A001110 (t is both triangular and square).
%C A238456 a(13) > 5*10^22. - _Giovanni Resta_, Mar 02 2014
%e A238456 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.
%t A238456 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 *)
%o A238456 (Python)
%o A238456 def isqrt(a):
%o A238456     sr = 1 << (int.bit_length(int(a)) >> 1)
%o A238456     while a < sr*sr:  sr>>=1
%o A238456     b = sr>>1
%o A238456     while b:
%o A238456         s = sr + b
%o A238456         if a >= s*s:  sr = s
%o A238456         b>>=1
%o A238456     return sr
%o A238456 t = i = 0
%o A238456 while 1:
%o A238456     t += i
%o A238456     i += 1
%o A238456     s = isqrt(t)
%o A238456     if s*s==t:  s-=1
%o A238456     txy = t + 2*s*(s+1) + 1   # t + s^2 + (s+1)^2
%o A238456     r = isqrt(txy)
%o A238456     if r*r==txy:  print(str(t), end=',')
%Y A238456 Cf. A000217, A000290, A001110, A238489.
%K A238456 nonn,hard,more
%O A238456 1,2
%A A238456 _Alex Ratushnyak_, Feb 26 2014
%E A238456 a(12) from _Giovanni Resta_, Mar 02 2014