A048724 -id:A048724 - cp's OEIS frontend

A242399 Write n and 3n in ternary representation and add all trits modulo 3.

0, 4, 8, 12, 16, 11, 24, 19, 23, 36, 40, 44, 48, 52, 47, 33, 28, 32, 72, 76, 80, 57, 61, 56, 69, 64, 68, 108, 112, 116, 120, 124, 119, 132, 127, 131, 144, 148, 152, 156, 160, 155, 141, 136, 140, 99, 103, 107, 84, 88, 83, 96, 91, 95, 216, 220, 224, 228, 232

Offset: 0

Reinhard Zumkeller, May 13 2014

			n = 25, 3*n = 75:
.  A007089(25) =  221
.  A007089(75) = 2210
.   add trits    ----
.    modulo 3    2101 = A007089(64), hence a(25) = 64.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
P. Mathonet, M. Rigo, M. Stipulanti and N. Zénaïdi, On digital sequences associated with Pascal's triangle, arXiv:2201.06636 [math.NT], 2022.
Eric Weisstein's World of Mathematics, Ternary.
Wikipedia, Ternary numeral system
Index entries for sequences related to carryless arithmetic

Cf. A004488, A007089, A008586, A030341, A048724, A242400, A242407.

Row / column 4 of A325820.

a242399 n = foldr (\t v -> 3 * v + t) 0 $
                  map (flip mod 3) $ zipWith (+) ([0] ++ ts) (ts ++ [0])
            where ts = a030341_row n

a(n) <= 4*n; a(m) = 4*m iff m is a term of A242407.

a(n) = A008586(n) - A242400(n).

A178731 a(n) = n XOR 5n, where XOR is bitwise XOR.

Original entry on oeis.org

0, 4, 8, 12, 16, 28, 24, 36, 32, 36, 56, 60, 48, 76, 72, 68, 64, 68, 72, 76, 112, 124, 120, 100, 96, 100, 152, 156, 144, 140, 136, 132, 128, 132, 136, 140, 144, 156, 152, 228, 224, 228, 248, 252, 240, 204, 200, 196, 192, 196, 200, 204, 304, 316, 312, 292, 288, 292

Offset: 0

Dmitry Kamenetsky, Jun 08 2010

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..5000

Cf. A048724, A178729, A048725, A178732, A178733, A178734, A178735, A178736. - Robert G. Wilson v, Jun 09 2010

[BitwiseXor(n, 5*n): n in [0..70]]; // Vincenzo Librandi, Jul 11 2017

f[n_] := BitXor[n, 5 n]; Array[f, 60, 0] (* Robert G. Wilson v, Jun 09 2010 *)

a(n) = bitxor(n, 5*n); \\ Michel Marcus, Jul 11 2017

a(30) onwards from Robert G. Wilson v, Jun 09 2010

A178734 a(n) = n XOR 8n, where XOR is bitwise XOR.

Original entry on oeis.org

0, 9, 18, 27, 36, 45, 54, 63, 72, 65, 90, 83, 108, 101, 126, 119, 144, 153, 130, 139, 180, 189, 166, 175, 216, 209, 202, 195, 252, 245, 238, 231, 288, 297, 306, 315, 260, 269, 278, 287, 360, 353, 378, 371, 332, 325, 350, 343, 432, 441, 418, 427, 404, 413, 390

Offset: 0

Dmitry Kamenetsky, Jun 08 2010

nonn

Cf. A048724, A178729, A048725, A178731, A178732, A178733, A178735, A178736. - Robert G. Wilson v, Jun 09 2010

f[n_] := BitXor[n, 8 n]; Array[f, 60, 0] (* Robert G. Wilson v, Jun 09 2010 *)

def A178734(n): return n^ n<<3 # Chai Wah Wu, Jun 29 2022

a(30) onwards from Robert G. Wilson v, Jun 09 2010

A234022 a(n) = A000120(A193231(n)); number of 1-bits in blue code for n.

Original entry on oeis.org

0, 1, 2, 1, 2, 1, 2, 3, 4, 3, 2, 3, 2, 3, 2, 1, 2, 1, 2, 3, 2, 3, 4, 3, 4, 5, 4, 3, 4, 3, 2, 3, 4, 3, 2, 3, 4, 5, 4, 3, 4, 5, 6, 5, 4, 3, 4, 5, 2, 3, 2, 1, 4, 3, 2, 3, 4, 3, 4, 5, 2, 3, 4, 3, 4, 3, 4, 5, 2, 3, 4, 3, 4, 5, 4, 3, 6, 5, 4, 5, 2, 3, 4, 3, 2, 1, 2

Offset: 0

Antti Karttunen, Dec 28 2013

nonn

Antti Karttunen, Table of n, a(n) for n = 0..8191
Joerg Arndt, Matters Computational (The Fxtbook), section 1.19 "Invertible transforms on words", pp. 49--55. [Cf. especially pages 50 & 51].

A234023 gives the positions where abs(a(n)-a(n+1)) > 1.

Cf. A000120, A193231.

def a065621(n): return n^(2*(n - (n&-n)))
def a048724(n): return n^(2*n)
l=[0, 1]
z=[0, 1]
for n in range(2, 101):
    if n%2==0: l.append(a048724(l[n//2]))
    else: l.append(a065621(1 + l[(n - 1)//2]))
    z.append(bin(l[-1])[2:].count("1"))
print(z) # Indranil Ghosh, Jun 05 2017

(define (A234022 n) (A000120 (A193231 n)))

a(n) = A000120(A193231(n)).

A000035(a(n)) = A000035(n) = (n mod 2) for all n. [Even terms occur only on even indices and odd terms only on odd indices, respectively]

A234025 Permutation of nonnegative integers: a(n) = A054429(A193231(n)).

Original entry on oeis.org

0, 1, 2, 3, 6, 7, 5, 4, 8, 9, 11, 10, 13, 12, 14, 15, 30, 31, 29, 28, 27, 26, 24, 25, 17, 16, 18, 19, 20, 21, 23, 22, 44, 45, 47, 46, 41, 40, 42, 43, 35, 34, 32, 33, 38, 39, 37, 36, 61, 60, 62, 63, 56, 57, 59, 58, 50, 51, 49, 48, 55, 54, 52, 53, 106, 107, 105

Offset: 0

Antti Karttunen, Dec 28 2013

nonn

Antti Karttunen, Table of n, a(n) for n = 0..8191
Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A234026.

def a065621(n): return n^(2*(n - (n&-n)))
def a048724(n): return n^(2*n)
def a054429(n): return 1 if n==1 else 2*a054429(int(n/2)) + 1 - n%2
def a193231(n):
    if n<2: return n
    if n%2==0: return a048724(a193231(n/2))
    else: return a065621(1 + a193231((n - 1)/2))
def a(n): return n if n<2 else a054429(a193231(n)) # Indranil Ghosh, Jun 05 2017

(define (A234025 n) (A054429 (A193231 n)))

a(n) = A054429(A193231(n)).

a(n) = A234027(A054429(n)).

A234612 Self-inverse permutation of nonnegative integers, "blue-gray" code: a(n) = A003188(A193231(n)).

Original entry on oeis.org

0, 1, 2, 3, 7, 6, 5, 4, 8, 9, 10, 11, 15, 14, 13, 12, 25, 24, 27, 26, 30, 31, 28, 29, 17, 16, 19, 18, 22, 23, 20, 21, 42, 43, 40, 41, 45, 44, 47, 46, 34, 35, 32, 33, 37, 36, 39, 38, 51, 50, 49, 48, 52, 53, 54, 55, 59, 58, 57, 56, 60, 61, 62, 63, 127, 126, 125

Offset: 0

Antti Karttunen, Dec 28 2013

nonn

Antti Karttunen, Table of n, a(n) for n = 0..8191
Joerg Arndt, Matters Computational (The Fxtbook), section 1.19 "Invertible transforms on words", pp. 49--55 [Cf. especially the identity 1.19-9d: gB = Bg^{-1} given on page 53].
Index entries for sequences that are permutations of the natural numbers

Cf. A003188, A006068, A193231, A234613, A234024-A234027.

def a065621(n): return n^(2*(n - (n&-n)))
def a048724(n): return n^(2*n)
def a003188(n): return n^(n>>1)
def a193231(n):
    if n<2: return n
    if n%2==0: return a048724(a193231(n/2))
    else: return a065621(1 + a193231((n - 1)/2))
def a(n): return n if n<2 else a003188(a193231(n)) # Indranil Ghosh, Jun 05 2017

(define (A234612 n) (A003188 (A193231 n)))
(define (A234612v2 n) (A193231 (A006068 n))) ;; Alternative 2.

a(n) = A003188(A193231(n)).
a(n) = A193231(A006068(n)).
a(n) = A193231(A234613(A193231(n))).

A136380 Quotient obtained when A036284(n) is considered as a GF(2)[X]-polynomial and it is divided by (x^3 + 1) ^ A000225(n-1).

Original entry on oeis.org

24, 160, 11968, 49657088, 837028380268032, 237269922100748727235760269312, 18811253173629696438994877569412700111469395859003555753984, 118178826602781220665226658680265194908312590801831513776333330179329649495708436476846379030238467286212637486694400

Offset: 1

Antti Karttunen, Dec 29 2007

A. Karttunen, Table of n, a(n) for n = 1..11
A. Karttunen, C program for computing this sequence

a(n) = 4*A136382(n) = 2*A048724(A136384(n)). A136381 shows the same sequence in octal base. Cf. A036284.

A136382 a(n) = A136380(n)/4.

Original entry on oeis.org

6, 40, 2992, 12414272, 209257095067008, 59317480525187181808940067328, 4702813293407424109748719392353175027867348964750888938496, 29544706650695305166306664670066298727078147700457878444083332544832412373927109119211594757559616821553159371673600

Offset: 1

Antti Karttunen, Dec 29 2007

Note that each term a(n) fits into A007283(n-1) bits.

A. Karttunen, Table of n, a(n) for n = 1..11
A. Karttunen, C program for computing this sequence

a(n) = A048724(A136384(n))/2. A136383 shows the same sequence in octal base. Cf. A036284.

A142149 a(n) = XOR{k OR (n-k): 0<=k<=n}.

Original entry on oeis.org

0, 1, 3, 3, 6, 5, 5, 7, 12, 9, 15, 11, 10, 13, 9, 15, 24, 17, 27, 19, 30, 21, 29, 23, 20, 25, 23, 27, 18, 29, 17, 31, 48, 33, 51, 35, 54, 37, 53, 39, 60, 41, 63, 43, 58, 45, 57, 47, 40, 49, 43, 51, 46, 53, 45, 55, 36, 57, 39, 59, 34, 61, 33, 63, 96, 65, 99, 67, 102, 69, 101, 71

Offset: 0

Reinhard Zumkeller, Jul 15 2008

a(n) = XOR{k AND (n-k): 0<=k<=n}.

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

Cf. A003817, A000004, A086099, A142150, A142151, A001477.

import Data.Bits (xor, (.|.))
a142149 :: Integer -> Integer
a142149 = foldl xor 0 . zipWith (.|.) [0..] . reverse . enumFromTo 1
-- Reinhard Zumkeller, Mar 31 2015

a(n)=if(n%2, n, bitxor(n, n/2)) \\ Charles R Greathouse IV, Jul 01 2022

def A142149(n): return n if n&1 else (n^ n>>1) # Chai Wah Wu, Jun 29 2022

a(2*n) = A048724(n) and a(2*n+1) = A005408(n).

A178732 a(n) = n XOR 6n, where XOR is bitwise XOR.

Original entry on oeis.org

0, 7, 14, 17, 28, 27, 34, 45, 56, 63, 54, 73, 68, 67, 90, 85, 112, 119, 126, 97, 108, 107, 146, 157, 136, 143, 134, 185, 180, 179, 170, 165, 224, 231, 238, 241, 252, 251, 194, 205, 216, 223, 214, 297, 292, 291, 314, 309, 272, 279, 286, 257, 268, 267, 370, 381

Offset: 0

Dmitry Kamenetsky, Jun 08 2010

nonn

Cf. A048724, A178729, A048725, A178731, A178733, A178734, A178735, A178736. - Robert G. Wilson v, Jun 09 2010

f[n_] := BitXor[n, 6 n]; Array[f, 60, 0] (* Robert G. Wilson v, Jun 09 2010 *)

a(30) onwards from Robert G. Wilson v, Jun 09 2010

cp's OEIS Frontend

A242399 Write n and 3n in ternary representation and add all trits modulo 3.

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A178731 a(n) = n XOR 5n, where XOR is bitwise XOR.

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A178734 a(n) = n XOR 8n, where XOR is bitwise XOR.

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A234022 a(n) = A000120(A193231(n)); number of 1-bits in blue code for n.

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A234025 Permutation of nonnegative integers: a(n) = A054429(A193231(n)).

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A234612 Self-inverse permutation of nonnegative integers, "blue-gray" code: a(n) = A003188(A193231(n)).

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A136380 Quotient obtained when A036284(n) is considered as a GF(2)[X]-polynomial and it is divided by (x^3 + 1) ^ A000225(n-1).

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A136382 a(n) = A136380(n)/4.

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A142149 a(n) = XOR{k OR (n-k): 0<=k<=n}.

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