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A367378 2*k-digit squares with the left half being a reversed k-digit square and the right half being a k-digit square.

%I A367378 #30 Jun 17 2025 00:39:39
%S A367378 49,1849,144400,148225,522729,16564900,40322500,46717225,98446084,
%T A367378 98863249,1429293636,1697440000,4013222500,4228250625,4247128900,
%U A367378 4684991809,5205622500,5227290000,6161465025,6557274529,104409765625,121975562500,123151864900,127413302500,140301186624
%N A367378 2*k-digit squares with the left half being a reversed k-digit square and the right half being a k-digit square.
%H A367378 Michael S. Branicky, <a href="/A367378/b367378.txt">Table of n, a(n) for n = 1..1031</a> (all terms <= 40 digits)
%H A367378 David A. Corneth, <a href="/A367378/a367378.gp.txt">PARI program</a>
%e A367378 522729 is in the sequence since reversed the left half is the square 15^2 and the right half is the square 27^2.
%e A367378 A 6-digit term might start with 522 as 522 is the reversal of a three-digit square (namely 225 = 15^2). If a 6-digit term starts with 522 then it is between (inclusive) 522100 and 522999. The only such square is 522729. As 729 (= 522729 - 522000) is a square we have 522729 is in the sequence. - _David A. Corneth_, Nov 21 2023
%o A367378 (Python)
%o A367378 from math import isqrt
%o A367378 from itertools import count, islice
%o A367378 def agen(): # generator of terms
%o A367378     for k in count(1):
%o A367378         lb, ub, sk = isqrt(10**(k-1)-1)+1, isqrt(10**k-1), set()
%o A367378         for i in range(lb, ub+1):
%o A367378             if i%10 == 0: continue
%o A367378             left = int(str(i*i)[::-1]) * 10**k
%o A367378             # loop below based on idea by _David A. Corneth_ in Example
%o A367378             lbt, ubt = isqrt(left-1)+1, isqrt(left + 10**k - 1)
%o A367378             for t in range(lbt, ubt+1):
%o A367378                 tt = t*t
%o A367378                 right = tt - left
%o A367378                 sr = str(right)
%o A367378                 if len(sr) == k and isqrt(right)**2 == right:
%o A367378                     sk.add(tt)
%o A367378         yield from sorted(sk)
%o A367378 print(list(islice(agen(), 20))) # _Michael S. Branicky_, Nov 21 2023
%o A367378 (PARI) \\ See PARI link
%Y A367378 Cf. A000290, A004086.
%K A367378 nonn,base
%O A367378 1,1
%A A367378 _Reiner Moewald_, Nov 15 2023
%E A367378 More terms from _David A. Corneth_, Nov 20 2023