A263609 Base-4 numbers whose square is a palindrome in base 4.

0, 1, 11, 101, 111, 1001, 1013, 1103, 10001, 10101, 10121, 10331, 100001, 100133, 1000001, 1001001, 1001201, 1010301, 1100211, 1100323, 1101211, 10000001, 10001333, 10013201, 10031113, 100000001, 100010001, 100012001, 100103001, 100301113, 100332101, 101002101, 103231203, 110002011

Offset

1,3

Links

Table of n, a(n) for n=1..34.

G. J. Simmons, On palindromic squares of non-palindromic numbers, J. Rec. Math., 5 (No. 1, 1972), 11-19. [Annotated scanned copy]

Example

From Mattew Bondar, Mar 12 2021: (Start)
111_4 = 21_10, 21^2 = 441, 441_10 = 12321_4 (palindrome).
1013_4 = 71_10, 71^2 = 5041, 5041_10 = 1032301_4 (palindrome). (End)

Mathematica

FromDigits /@ IntegerDigits[Select[Range[0, 2^17], PalindromeQ@ IntegerDigits[#^2, 4] &], 4] (* Michael De Vlieger, Mar 13 2021 *)

Prog

(Python)
def decimal_to_quaternary(n):
    if n == 0:
        return '0'
    b = ''
    while n > 0:
        b = str(n % 4) + b
        n = n // 4
    return b
x = 0
counter = 0
while True:
    y = decimal_to_quaternary(x ** 2)
    if y == y[::-1]:
        print(int(decimal_to_quaternary(x)))
        counter += 1
    x += 1  # Mattew Bondar, Mar 10 2021

Crossrefs

Cf. A002778, A029986, A263610, A029987.

Sequence in context: A265528 A099821 A193415 * A333415 A264406 A057148

Adjacent sequences: A263606 A263607 A263608 * A263610 A263611 A263612

Keyword

nonn,base

Author

N. J. A. Sloane, Oct 22 2015

Status

approved