A055948 -id:A055948 - cp's OEIS frontend

A055944 a(n) = n + (reversal of base-2 digits of n) (written in base 10).

0, 2, 3, 6, 5, 10, 9, 14, 9, 18, 15, 24, 15, 24, 21, 30, 17, 34, 27, 44, 25, 42, 35, 52, 27, 44, 37, 54, 35, 52, 45, 62, 33, 66, 51, 84, 45, 78, 63, 96, 45, 78, 63, 96, 57, 90, 75, 108, 51, 84, 69, 102, 63, 96, 81, 114, 63, 96, 81, 114, 75, 108, 93, 126, 65, 130, 99, 164, 85

Offset: 0

Henry Bottomley, Jul 18 2000

If n has an even number of digits in base-2 then a(n) is a multiple of 3.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A030101, A035522 (iterated), A055948.

a055944 n = n + a030101 n  -- Reinhard Zumkeller, Nov 14 2011

f[n_] := Block[{id = IntegerDigits[n, 2]}, FromDigits[id + Reverse@id, 2]]; Array[f, 69, 0] (* Robert G. Wilson v, Nov 07 2010 *)
Array[#+FromDigits[Reverse[IntegerDigits[#,2]],2]&,70,0] (* Harvey P. Dale, Feb 09 2025 *)

a(n) = n + A030101(n).

A030103 Base 4 reversal of n (written in base 10).

Original entry on oeis.org

0, 1, 2, 3, 1, 5, 9, 13, 2, 6, 10, 14, 3, 7, 11, 15, 1, 17, 33, 49, 5, 21, 37, 53, 9, 25, 41, 57, 13, 29, 45, 61, 2, 18, 34, 50, 6, 22, 38, 54, 10, 26, 42, 58, 14, 30, 46, 62, 3, 19, 35, 51, 7, 23, 39, 55, 11, 27, 43, 59, 15, 31, 47, 63, 1, 65, 129, 193, 17, 81, 145, 209, 33, 97, 161

Offset: 0

David W. Wilson

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A004086, A030101 - A030108, A048703, A055948, A035524.

import Data.List (unfoldr)
a030103 n = foldl (\v d -> 4*v + d) 0 $ unfoldr dig n where
    dig x = if x == 0 then Nothing else Just $ swap $ divMod x 4
-- Reinhard Zumkeller, Oct 10 2011

IntegerReverse[Range[0, 100], 4] (* Paolo Xausa, Aug 07 2024 *)

a(n,b=4)=subst(Polrev(base(n,b)),x,b) /* where */
base(n,b)={my(a=[n%b]);while(0M. F. Hasler, Nov 04 2011
(MIT/GNU Scheme)
(define (A030103 n) (if (zero? n) n (let ((uplim (+ (A000523 n) (- 1 (modulo (A000523 n) 2))))) (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)))))))
;; Antti Karttunen, Oct 30 2013

A035524 Reverse and add (in base 4).

Original entry on oeis.org

1, 2, 4, 5, 10, 20, 25, 50, 85, 170, 340, 425, 850, 1385, 3070, 6140, 10225, 15335, 29410, 65135, 129070, 317675, 1280860, 2163725, 3999775, 7999550, 20321515, 81946460, 138412045, 255852575, 511705150, 1300234475, 5242880860

Offset: 0

N. J. A. Sloane

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Index entries for sequences related to Reverse and Add!

Cf. A035522.

Cf. A030103.

a035524 n = a035524_list !! n
a035524_list = iterate a055948 1
-- Reinhard Zumkeller, Oct 10 2011

nxt4[n_]:=Module[{idn4=IntegerDigits[n,4]},FromDigits[idn4+ Reverse[idn4], 4]]; NestList[nxt4,1,40] (* Harvey P. Dale, May 02 2011 *)

def reversedigits(n, b=10): # reverse digits of n in base b
    x, y = n, 0
    while x >= b:
        x, r = divmod(x, b)
        y = b*y + r
    return b*y + x
A035524_list, l = [1], 1
for _ in range(50):
    l += reversedigits(l,4)
    A035524_list.append(l)

a(n+1) = A055948(a(n)), a(0) = 1. [Reinhard Zumkeller, Oct 10 2011]

More terms from Larry Reeves (larryr(AT)acm.org), Sep 22 2000

A319804 a(n) = A319720(n) + A319652(n).

Original entry on oeis.org

0, 2, 4, 6, 5, 10, 15, 20, 10, 15, 20, 25, 15, 20, 25, 30, 17, 25, 42, 59, 25, 42, 59, 76, 42, 59, 67, 84, 59, 76, 84, 92, 34, 42, 50, 67, 42, 59, 67, 84, 50, 67, 84, 101, 67, 84, 101, 109, 51, 59, 67, 75, 59, 76, 84, 92, 67, 84, 101, 109, 75, 92, 109, 126, 65, 85, 150, 215

Offset: 0

Seiichi Manyama, Sep 28 2018

This sequence is different from A055948.

Seiichi Manyama, Table of n, a(n) for n = 0..4095

Base b: A319785 (b=2), A319803 (b=3), this sequence (b=4), A319805 (b=5), A319806 (b=6), A319807 (b=7), A319808 (b=8), A319747 (b=9), A052008 (b=10).

Cf. A055948, A319652, A319720.

Table[FromDigits[Reverse[#], 4] + FromDigits[#, 4] & [Sort[IntegerDigits[n, 4]]], {n, 0, 100}] (* Paolo Xausa, Aug 07 2024 *)

a(n) = my(nd=digits(n, 4)); fromdigits(vecsort(nd), 4) + fromdigits(vecsort(nd,,4), 4); \\ Michel Marcus, Sep 28 2018

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

Original entry on oeis.org

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

Antti Karttunen, Mar 07 1999

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.

			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.

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

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.

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:

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

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

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

cp's OEIS Frontend

A055944 a(n) = n + (reversal of base-2 digits of n) (written in base 10).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

Formula

A030103 Base 4 reversal of n (written in base 10).

Views

Author

Keywords

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

A035524 Reverse and add (in base 4).

Views

Author

Keywords

Links

Crossrefs

Programs

Haskell

Mathematica

Python

Formula

Extensions

A319804 a(n) = A319720(n) + A319652(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Python

Formula