A048703 Numbers which in base 4 are palindromes and have an even number of digits.

0, 5, 10, 15, 65, 85, 105, 125, 130, 150, 170, 190, 195, 215, 235, 255, 1025, 1105, 1185, 1265, 1285, 1365, 1445, 1525, 1545, 1625, 1705, 1785, 1805, 1885, 1965, 2045, 2050, 2130, 2210, 2290, 2310, 2390

Offset: 0

Author

Antti Karttunen, Mar 07 1999

Keywords

• nonn
• base

Comments

In quaternary base (base 4) the terms look like 0, 11, 22, 33, 1001, 1111, 1221, 1331, 2002, 2112, 2222, 2332, 3003, 3113, 3223, 3333, 100001, 101101, 102201, ..., which is a subsequence of A118595.

Zero is included as a(0) because we can consider it as having zero digits after leading zeros have been excluded.

Examples

			Each a(n) is obtained by concatenating the original base-4 expansion of n (which comes to the left hand, i.e., the most significant side) with its mirror-image (which comes to the right hand, i.e., the least significant side). For example, for a(4) we have 4 = '10' in base 4, which concatenated with its reversal '01' yields '1001', which when converted back to decimal yields 1*64 + 0*16 + 0*4 + 1*1 = 65, thus a(4)=65.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 0..10000 (terms 0..4095 from Antti Karttunen)

Crossrefs

Subsequence of A014192 (all numbers which are palindromes in base 4, including also those of odd number of digits).

Cf. also A048704 (this sequence divided by 5), A048701 (binary palindromes of even length), A055948, A110591, A118595, A030103, A007090, A000302.

Programs

Maple

A048703(n) := (n) -> (2^(floor_log_2_coarse(n)+1))*n + sum('(bit_i(n, i+((-1)^i))*(2^(floor_log_2_coarse(n)-i)))', 'i'=0..floor_log_2_coarse(n));
bit_i := (x,i) -> `mod`(floor(x/(2^i)),2);
# Following is like floor_log_2 but even results are incremented by one:
floor_log_2_coarse := proc(n) local nn,i: nn := n; for i from -1 to n do if(0 = nn) then RETURN(i+(1-(i mod 2))); fi: nn := floor(nn/2); od: end:

Mathematica

q[n_] := EvenQ[IntegerLength[n, 4]] && PalindromeQ[IntegerDigits[n, 4]]; Select[Range[0, 2400, 5], q] (* Amiram Eldar, May 27 2024 *)

Python

def A048703(n):
    s = bin(n-1)[2:]
    if len(s) % 2: s = '0'+s
    t = [s[i:i+2] for i in range(0,len(s),2)]
    return int(''.join(t+t[::-1]),2) # Chai Wah Wu, Feb 26 2021

Formula

a(0) = 0, and for n >= 1, a(n) = A030103(n) + (n * A000302(A110591(n))). - Antti Karttunen, Oct 30 2013

a(n) = 5*A048704(n). [This is just a consequence of the definition of A048704.] - Antti Karttunen, Jul 25 2013