A246159 -id:A246159 - cp's OEIS frontend

A065620 a(0)=0; thereafter a(2n) = 2a(n), a(2n+1) = -2a(n) + 1.

0, 1, 2, -1, 4, -3, -2, 3, 8, -7, -6, 7, -4, 5, 6, -5, 16, -15, -14, 15, -12, 13, 14, -13, -8, 9, 10, -9, 12, -11, -10, 11, 32, -31, -30, 31, -28, 29, 30, -29, -24, 25, 26, -25, 28, -27, -26, 27, -16, 17, 18, -17, 20, -19, -18, 19, 24, -23, -22, 23, -20, 21, 22, -21, 64, -63, -62, 63, -60, 61, 62, -61, -56

Offset: 0

Marc LeBrun, Nov 07 2001

Reversing binary value of n: convert sum of powers of 2 in binary representation of n to alternating sum.

The alternation is applied only to the nonzero bits and does not depend on the exponent of 2. All integers have a unique reversing binary representation (see cited Knuth exercise for proof).

			11 = 1 + 2 + 8 -> 1 - 2 + 8 = 7 = a(11).

Donald E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, 1969, Vol. 2, p. 178 (exercise 4.1. Nr. 27).

N. J. A. Sloane, Table of n, a(n) for n = 0..20000 (First 8192 terms from Reinhard Zumkeller)

Cf. A010060, A048724, A065621, A246159, A246160, A265263.

The negative of entry A104895.

import Data.List (transpose)
a065620 n = a065620_list !! n
a065620_list = 1 : concat (transpose [zs, map ((+ 1) . negate) zs])
               where zs = map (* 2) a065620_list
-- Reinhard Zumkeller, Mar 26 2014

f:=proc(n) option remember; if n=0 then 0 elif (n mod 2) = 0 then 2*f(n/2) else 1-2*f((n-1)/2); fi; end;
[seq(f(n),n=0..100)]; # N. J. A. Sloane, Apr 25 2018

a[0] = 0; a[n_]:= a[n]= If[OddQ[n], 1 - 2*a[(n-1)/2], 2*a[n/2]]; Array[a, 100, 0] (* Amiram Eldar, Sep 05 2023 *)

A065620(n,c=1)=sum(i=0,logint(n+!n,2),if(bittest(n,i),(-1)^c++<M. F. Hasler, Apr 16 2018

def a(n): return n if n<3 else 2*a(n/2) if n%2==0 else -2*a((n - 1)/2) + 1 # Indranil Ghosh, Jun 07 2017

def A065620(n):
    c, a, b = 0, -1, 1
    for j in bin(n)[-1:1:-1]:
        if int(j):
            c += (a:=-a)*b
        b <<= 1
    return c # Chai Wah Wu, Sep 19 2023

(definec (A065620 n) (cond ((zero? n) n) ((even? n) (* 2 (A065620 (/ n 2)))) (else (+ 1 (* -2 (A065620 (/ (- n 1) 2)))))))
;; using memoization-macro definec from IntSeq-library of Antti Karttunen, Aug 18 2014

a(n) = if n<2 then n else b+2*(1-2*b)*a((n-b)/2) where b is the least significant bit in n.
From Antti Karttunen, Aug 18 2014: (Start)
a(n) = A246160(n) - A246159(n).
a(A065621(n)) = n.
a(A048724(n)) = -n. (End)
a(n) = -A104895(n). - M. F. Hasler, Apr 16 2018
a(n) = (2*A265263(n) - n) * (2*A010060(n) - 1). - Alan Michael Gómez Calderón, Jul 01 2025

Formula fixed by Reinhard Zumkeller, Mar 26 2014
Extended to a(0) = 0 by M. F. Hasler, Apr 16 2018
Edited by N. J. A. Sloane, Apr 25 2018

A245710 Number of nonzero evil numbers <= n, see A001969.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 3, 3, 3, 4, 5, 5, 6, 6, 6, 7, 7, 8, 9, 9, 10, 10, 10, 11, 12, 12, 12, 13, 13, 14, 15, 15, 15, 16, 17, 17, 18, 18, 18, 19, 20, 20, 20, 21, 21, 22, 23, 23, 24, 24, 24, 25, 25, 26, 27, 27, 27, 28, 29, 29, 30, 30, 30, 31, 31, 32, 33, 33, 34, 34, 34, 35, 36

Offset: 0

Antti Karttunen, Aug 18 2014

nonn

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 52.

One less than A159481.

Cf. A001969, A010059, A115384, A246159, A246162.

Join[{0},Accumulate[Table[If[EvenQ[DigitCount[n,2,1]],1,0],{n,80}]]] (* Harvey P. Dale, Aug 01 2021 *)

def A245710(n): return (n+1>>1)-((n+1).bit_count()&1&(n+1)^1) # Chai Wah Wu, Mar 01 2023

a(0) = 0, and for n >= 1, a(n) = A010059(n) + a(n-1). [Partial sums of A010059, after ignoring the first one at zero].
a(n) = n  - A115384(n).
a(n) = A159481(n)-1.

A246164 Permutation of natural numbers: a(1) = 1, a(A065621(n)) = A014580(a(n-1)), a(A048724(n)) = A091242(a(n)), where A065621(n) and A048724(n) are the reversing binary representation of n and -n, respectively, and A014580 resp. A091242 are the binary coded irreducible resp. reducible polynomials over GF(2).

Original entry on oeis.org

1, 2, 4, 11, 8, 5, 3, 7, 6, 9, 13, 17, 47, 31, 14, 61, 21, 42, 185, 24, 87, 319, 62, 12, 25, 19, 10, 59, 20, 15, 37, 229, 49, 22, 67, 76, 415, 103, 28, 18, 55, 137, 34, 41, 16, 27, 97, 78, 425, 109, 29, 1627, 222, 54, 283, 433, 79, 373, 3053, 33, 131, 647, 108, 847, 133, 745, 6943, 44, 193, 1053, 160, 504, 4333, 587, 99

Offset: 1

Antti Karttunen, Aug 19 2014

nonn

This is an instance of entanglement permutation, where the two complementary pairs to be entangled with each other are A065621/A048724 and A014580/A091242 (binary codes for irreducible and reducible polynomials over GF(2)).
The former are themselves permutations of A000069/A001969 (odious and evil numbers), which means that this permutation shares many properties with A246162.
For the comments about the cycle structure, please see A246163.

Inverse: A246163.
Similar or related permutations: A246206, A246202, A193231, A245702, A234025, A246162, A234612, A246204.
Cf. A010060, A014580, A048724, A065620, A065621, A091242, A246159, A246160.

a(1) = 1, and for n > 1, if A010060(n) = 1 [i.e. when n is an odious number], a(n) = A014580(a(A065620(n)-1)), otherwise a(n) = A091242(a(- (A065620(n)))). [A065620 Converts sum of powers of 2 in binary representation of n to an alternating sum].
As a composition of related permutations:
a(n) = A246202(A193231(n)).
a(n) = A245702(A234025(n)).
a(n) = A246162(A234612(n)).
a(n) = A193231(A246204(A193231(n))).
For all n > 1, A091225(a(n)) = A010060(n). [Maps odious numbers to binary representations of irreducible GF(2) polynomials (A014580) and evil numbers to the corresponding representations of reducible polynomials (A091242), in some order. A246162 has the same property].

A246160 Inverse function to the injection A065621.

Original entry on oeis.org

0, 1, 2, 0, 4, 0, 0, 3, 8, 0, 0, 7, 0, 5, 6, 0, 16, 0, 0, 15, 0, 13, 14, 0, 0, 9, 10, 0, 12, 0, 0, 11, 32, 0, 0, 31, 0, 29, 30, 0, 0, 25, 26, 0, 28, 0, 0, 27, 0, 17, 18, 0, 20, 0, 0, 19, 24, 0, 0, 23, 0, 21, 22, 0, 64, 0, 0, 63, 0, 61, 62, 0, 0, 57, 58, 0, 60, 0, 0, 59, 0, 49, 50, 0, 52

Offset: 0

Antti Karttunen, Aug 18 2014

nonn

Sequence has nonzero values a(n) = k at those points n for which A065621(k) = n and zeros at those positions n which are not present in A065621.

Equally, sequence is obtained when the negative terms of A065620 are replaced with zeros

Antti Karttunen, Table of n, a(n) for n = 0..8192

Cf. A246159, A006068, A010060, A115384, A065620, A065621.

a065620(n) = if(n<3, n, if(n%2, -2*a065620((n - 1)/2) + 1, 2*a065620(n/2)));
a(n) = (hammingweight(n)%2)*a065620(n);
for(n=0, 100, print1(a(n),", ")) \\ Indranil Ghosh, Jun 07 2017

def a065620(n): return n if n<3 else 2*a065620(n//2) if n%2==0 else -2*a065620((n - 1)//2) + 1
def a(n): return (bin(n)[2:].count("1")%2)*a065620(n)
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 07 2017

a(n) = A010060(n) * A065620(n).
a(n) = A246159(n) + A065620(n).
a(0) = 0, and for n >= 1, a(n) = A010060(n) *  (1 + A006068(A115384(n)-1)).
For all n, a(A065621(n)) = n.

A245812 Self-inverse permutation of natural numbers: a(0) = 0, a(1) = 1, and for n > 1, if A065620(n) < 0, a(n) = A065621(1+a(-(A065620(n)))), otherwise a(n) = A048724(a(A065620(n)-1)).

Original entry on oeis.org

0, 1, 3, 2, 6, 7, 4, 5, 15, 14, 13, 12, 11, 10, 9, 8, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23, 57, 56, 59, 58, 61, 60, 63, 62, 49, 48, 51, 50, 53, 52, 55, 54, 41, 40, 43, 42, 45, 44, 47, 46, 33, 32, 35, 34, 37, 36, 39, 38, 106, 107, 104, 105, 110, 111, 108, 109, 98, 99, 96, 97, 102, 103, 100

Offset: 0

Antti Karttunen, Aug 20 2014

nonn

This is an instance of entanglement permutation, where complementary pair A048724/A065621 is entangled with the same pair in the opposite order: A065621/A048724, with a(1) set to 1.

Note how this is A193231-conjugate of A054429.

Similar or related permutations: A193231, A234025, A234026, A246211, A236854, A235491, A054429, A234027.

Cf. A010060, A048724, A065620, A065621, A246159, A246160.

a048724(n) = bitxor(n, 2*n);
a065620(n) = if(n<3, n, if(n%2, -2*a065620((n - 1)/2) + 1, 2*a065620(n/2)));
a065621(n) = bitxor(n, 2*(n - bitand(n, -n)));
a(n) = x=a065620(n); if(n<2, n, if(x<0, a065621(1 + a(-x)), a048724(a(x - 1))));
for(n=0, 100, print1(a(n),", ")) \\ Indranil Ghosh, Jun 07 2017

def a048724(n): return n^(2*n)
def a065620(n): return n if n<3 else 2*a065620(n//2) if n%2==0 else -2*a065620((n - 1)//2) + 1
def a065621(n): return n^(2*(n - (n & -n)))
def a(n):
    x=a065620(n)
    return n if n<2 else a065621(1 + a(-x)) if x<0 else a048724(a(x - 1))
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 07 2017

a(0) = 0, a(1) = 1, and for n > 1, if A065620(n) < 0, a(n) = A065621(1+a(-(A065620(n)))), otherwise a(n) = A048724(a(A065620(n)-1)).
Equally:
a(0) = 0, a(1) = 1, and for n > 1, if A010060(n) = 0, a(n) = A065621(1+a(A246159(n))), otherwise a(n) = A048724(a(A246160(n)-1)). [Note how A246159 is an inverse function for A048724, while A246160 is an inverse function for A065621].
As a composition of related permutations:
a(n) = A193231(A234025(n)).
a(n) = A234026(A193231(n)).
a(n) = A193231(A054429(A193231(n))).

cp's OEIS Frontend

A065620 a(0)=0; thereafter a(2n) = 2a(n), a(2n+1) = -2a(n) + 1.

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A245710 Number of nonzero evil numbers <= n, see A001969.

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A246160 Inverse function to the injection A065621.

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A245812 Self-inverse permutation of natural numbers: a(0) = 0, a(1) = 1, and for n > 1, if A065620(n) < 0, a(n) = A065621(1+a(-(A065620(n)))), otherwise a(n) = A048724(a(A065620(n)-1)).

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