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

49, 1849, 144400, 148225, 522729, 16564900, 40322500, 46717225, 98446084, 98863249, 1429293636, 1697440000, 4013222500, 4228250625, 4247128900, 4684991809, 5205622500, 5227290000, 6161465025, 6557274529, 104409765625, 121975562500, 123151864900, 127413302500, 140301186624

Offset

1,1

Links

Michael S. Branicky, Table of n, a(n) for n = 1..1031 (all terms <= 40 digits)

David A. Corneth, PARI program

Example

522729 is in the sequence since reversed the left half is the square 15^2 and the right half is the square 27^2.
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

Prog

(Python)
from math import isqrt
from itertools import count, islice
def agen(): # generator of terms
    for k in count(1):
        lb, ub, sk = isqrt(10**(k-1)-1)+1, isqrt(10**k-1), set()
        for i in range(lb, ub+1):
            if i%10 == 0: continue
            left = int(str(i*i)[::-1]) * 10**k
            # loop below based on idea by David A. Corneth in Example
            lbt, ubt = isqrt(left-1)+1, isqrt(left + 10**k - 1)
            for t in range(lbt, ubt+1):
                tt = t*t
                right = tt - left
                sr = str(right)
                if len(sr) == k and isqrt(right)**2 == right:
                    sk.add(tt)
        found += len(sk)
        yield from sorted(sk)
print(list(islice(agen(), 20))) # Michael S. Branicky, Nov 21 2023
(PARI) \\ See PARI link

Crossrefs

Cf. A000290, A004086.

Sequence in context: A145848 A014942 A260856 * A065785 A163927 A245036

Adjacent sequences: A367375 A367376 A367377 * A367379 A367380 A367381

Keyword

nonn,base

Author

Reiner Moewald, Nov 15 2023

Extensions

More terms from David A. Corneth, Nov 20 2023

Status

approved