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

0,2

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.

Links

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

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

Example

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.

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 *)

Prog

(Scheme)
(define (A048703 n) (if (zero? n) n (let ((uplim (+ (A000523 n) (- 1 (modulo (A000523 n) 2))))) (+ (* (expt 2 (+ 1 uplim)) n) (add (lambda (i) (* (bit_i n (+ i (expt -1 i))) (expt 2 (- uplim i)))) 0 uplim)))))
(define (bit_i n i) (modulo (floor->exact (/ n (expt 2 i))) 2))
;; The functional add implements sum_{i=lowlim..uplim} intfun(i):
(define (add intfun lowlim uplim) (let sumloop ((i lowlim) (res 0)) (cond ((> i uplim) res) (else (sumloop (1+ i) (+ res (intfun i)))))))
;; Another version based on using A030103:
(define (A048703v2 n) (if (zero? n) n (+ (* (A000302 (A110591 n)) n) (A030103 n))))
(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

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.

Sequence in context: A022600 A067237 A196157 * A020332 A244025 A044830

Adjacent sequences: A048700 A048701 A048702 * A048704 A048705 A048706

Keyword

nonn,base

Author

Antti Karttunen, Mar 07 1999

Status

approved