A006068 -id:A006068 - cp's OEIS frontend

A268670 a(n) = A006068(A268669(n)).

1, 3, 1, 7, 1, 3, 5, 15, 5, 3, 13, 7, 9, 11, 1, 31, 1, 11, 29, 7, 25, 27, 9, 15, 17, 19, 5, 23, 13, 3, 21, 63, 21, 3, 61, 23, 57, 59, 13, 15, 49, 51, 17, 55, 5, 19, 53, 31, 33, 35, 1, 39, 29, 11, 37, 47, 9, 27, 45, 7, 41, 43, 25, 127, 25, 43, 125, 7, 121, 123, 41, 47, 113, 115, 9, 119, 45, 27, 117, 31, 97, 99, 33, 103, 1

Offset: 1

Antti Karttunen, Feb 10 2016

All terms are odd, by definition.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A001317 (positions of ones).

Cf. A006068, A268669, A268671.

f[n_] := Fold[BitXor, n, Quotient[n, 2^Range[BitLength@ n - 1]]]; g[n_] := If[OddQ@ #, n, g[#/2]] &@ f[n]; Table[f@ g@ n, {n, 85}]  (* Michael De Vlieger, Feb 12 2016, after Jan Mangaldan at A006068 *)

(define (A268670 n) (A006068 (A268669 n)))

a(n) = A006068(A268669(n)).

A268716 a(n) = 2*A006068(n); main diagonal of A268714.

Original entry on oeis.org

0, 2, 6, 4, 14, 12, 8, 10, 30, 28, 24, 26, 16, 18, 22, 20, 62, 60, 56, 58, 48, 50, 54, 52, 32, 34, 38, 36, 46, 44, 40, 42, 126, 124, 120, 122, 112, 114, 118, 116, 96, 98, 102, 100, 110, 108, 104, 106, 64, 66, 70, 68, 78, 76, 72, 74, 94, 92, 88, 90, 80, 82, 86, 84, 254, 252, 248, 250, 240, 242, 246, 244, 224, 226

Offset: 0

Antti Karttunen, Feb 12 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 0..8191

Cf. A001969, A006068.
Main diagonal of array A268714.
Row 3 and column 3 of array A268724.

def A268716(n):
    k, m = n, n>>1
    while m > 0:
        k ^= m
        m >>= 1
    return k<<1 # Chai Wah Wu, Jun 30 2022

Scheme
```
(define (A268716 n) (* 2 (A006068 n)))
```

a(n) = 2*A006068(n).
a(n) = A006068(A001969(n+1)).
a(n) = A268714(n,n).

A277902 If A010060(n) = 1, a(n) = A000069(A268671(n)), otherwise a(n) = A001969(1+a(A006068(n)/2)).

Original entry on oeis.org

1, 2, 3, 7, 6, 5, 4, 14, 9, 10, 13, 15, 8, 11, 12, 31, 24, 23, 28, 30, 25, 26, 17, 29, 16, 19, 18, 22, 27, 20, 21, 62, 43, 40, 61, 45, 56, 59, 54, 58, 49, 50, 33, 55, 36, 39, 52, 63, 32, 35, 48, 38, 57, 46, 37, 47, 34, 53, 44, 60, 41, 42, 51, 127, 102, 85, 124, 120, 121, 122, 83, 95, 112, 115, 68, 118, 89, 106

Offset: 1

Antti Karttunen, Nov 03 2016

a(n) gives the number that is in the same position in array A277880 as where n is located in array A277820.

			The top left corner of array A277820 is:
   1,  3,  5, 15
   2,  6, 10, 30
   7,  9, 27, 45
   4, 12, 20, 60
  13, 23, 57, 75
while the top left corner of A277880 is:
   1,  3,  6, 12
   2,  5, 10, 20
   4,  9, 18, 36
   7, 15, 30, 60
   8, 17, 34, 68
thus for example, a(1) = 1, a(2) = 2, a(3) = 3, a(4) = 7, a(5) = 6, a(6) = 5, a(7) = 4, a(9) = 9, a(12) = 15, a(13) = 8 and a(27) = 18.

Inverse: A277901.
Cf. A000069, A001969, A001317, A003945, A006068, A010060, A065621, A129771, A268671, A277823, A277825.
Related permutations and arrays: A277820, A277821, A277880.

If A010060(n) = 1 [when n is one of the odious numbers, A000069], then a(n) = A000069(A268671(n)), otherwise a(n) = A001969(1+a(A006068(n)/2)).
As a composition of other permutations:
a(n) = A277880(A277821(n)).
Other identities. For all n >= 1:
A010060(a(n)) = A010060(n). [Preserves the parity of binary weight.]
a(A001317(n)) = A003945(n).
a(A065621(n)) = A000069(n).
a(A277823(n)) = A129771(n).
a(A277825(n)) = 2*A129771(n).

A297110 Xor-Moebius transform of A006068, inverse of the binary Gray code.

Original entry on oeis.org

1, 2, 3, 4, 7, 4, 4, 8, 12, 8, 12, 8, 8, 12, 15, 16, 31, 20, 28, 16, 31, 20, 27, 16, 23, 24, 28, 24, 23, 16, 20, 32, 48, 32, 63, 40, 56, 36, 48, 32, 48, 32, 51, 40, 48, 44, 52, 32, 36, 56, 63, 48, 39, 36, 47, 48, 48, 56, 44, 32, 40, 60, 63, 64, 112, 80, 124, 64, 96, 64, 123, 80, 112, 72, 111, 72, 127, 80, 116, 64

Offset: 1

Antti Karttunen, Dec 25 2017

Unique sequence satisfying SumXOR_{d divides n} a(d) = A006068(n) for all n > 0, where SumXOR is the analog of summation under the binary XOR operation. See A295901 for a list of some of the properties of the Xor-Moebius transform.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to binary expansion of n

Cf. A006068, A256739, A295901, A296208.

A006068(n)= { my(s=1, ns); while(1, ns = n >> s; if(0==ns, break()); n = bitxor(n, ns); s <<= 1; ); return (n); } \\ Essentially Joerg Arndt's Jul 19 2012 code.
A297110(n) = { my(v=0); fordiv(n, d, if(issquarefree(n/d), v=bitxor(v, A006068(d)))); (v); };

A324337 a(n) = A002487(A006068(n)).

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 1, 3, 4, 3, 2, 5, 1, 4, 5, 3, 5, 4, 3, 7, 2, 7, 8, 5, 1, 5, 7, 4, 7, 5, 3, 8, 6, 5, 4, 9, 3, 10, 11, 7, 2, 9, 12, 7, 11, 8, 5, 13, 1, 6, 9, 5, 10, 7, 4, 11, 9, 7, 5, 12, 3, 11, 13, 8, 7, 6, 5, 11, 4, 13, 14, 9, 3, 13, 17, 10, 15, 11, 7, 18, 2, 11, 16, 9, 17, 12, 7, 19, 14, 11, 8, 19, 5, 18, 21, 13, 1, 7, 11, 6, 13, 9, 5, 14, 13, 10

Offset: 0

Antti Karttunen, Feb 23 2019

nonn

Like in A324338, a few terms preceding each position n = 2^k seem to be a batch of nearby Fibonacci numbers in some order.

For all n > 0 A324338(n)/A324337(n) constitutes an enumeration system of all positive rationals. For all n > 0 A324338(n) + A324337(n) = A071585(n). - Yosu Yurramendi, Oct 22 2019

Antti Karttunen, Table of n, a(n) for n = 0..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65537
Index entries for sequences related to binary expansion of n
Index entries for sequences related to Stern's sequences

Cf. A002487, A006068, A063946, A071585, A248663, A283477, A324287, A324338.

Block[{f}, f[m_] := Module[{a = 1, b = 0, n = m}, While[n > 0, If[OddQ[n], b = a + b, a = a + b]; n = Floor[n/2]]; b]; Array[f@ Fold[BitXor, #, Quotient[#, 2^Range[BitLength[#] - 1]]] &, 106, 0]] (* Michael De Vlieger, Dec 14 2019, after Jean-François Alcover at A002487 and Jan Mangaldan at A006068 *)

A006068(n)= { my(s=1, ns); while(1, ns = n >> s; if(0==ns, break()); n = bitxor(n, ns); s <<= 1; ); return (n); } \\ From A006068
A002487(n) = { my(s=sign(n), a=1, b=0); n = abs(n); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (s*b); };
A324337(n) = A002487(A006068(n));

maxlevel <- 6 # by choice
#
b <- 0; A324338 <- 1; A324337 <- 1
for(i in 1:2^maxlevel) {
  b[2*i  ] <-     b[i]
  b[2*i+1] <- 1 - b[i]
  A324338[2*i  ] <- A324338[i]          + A324337[i]*   b[i]
  A324338[2*i+1] <- A324338[i]          + A324337[i]*(1-b[i])
  A324337[2*i  ] <- A324338[i]*(1-b[i]) + A324337[i]
  A324337[2*i+1] <- A324338[i]*   b[i]  + A324337[i]}
#
A324338[1:127]; A324337[1:127]
# Yosu Yurramendi, Oct 22 2019

From Yosu Yurramendi, Oct 22 2019: (Start)
a(2^m+        k) = A324338(2^m+2^(m-1)+k), m > 0, 0 <= k < 2^(m-1)
a(2^m+2^(m-1)+k) = A324338(2^m+        k), m > 0, 0 <= k < 2^(m-1). (End)
a(n) = A324338(A063946(n)), n > 0. Yosu Yurramendi, Nov 04 2019
a(n) = A002487(A248663(A283477(n))). - Antti Karttunen, Nov 06 2019
a(n) = A002487(1+A233279(n)). - Yosu Yurramendi, Nov 08 2019
From Yosu Yurramendi, Nov 28 2019: (Start)
a(2^(m+1)+k) - a(2^m+k) = A324338(k), m >= 0, 0 <= k < 2^m.
a(A059893(2^(m+1)+A001969(k+1))) - a(A059893(2^m+A001969(k+1))) = A071585(k), m >= 0, 0 <= k < 2^(m-1).
a(A059893(2^(m+1)+ A000069(k+1))) = A071585(k), m >= 1, 0 <= k < 2^(m-1). (End)
From Yosu Yurramendi, Nov 29 2019: (Start)
For n > 0:
A324338(n) + A324337(n) = A071585(n).
A324338(2*A001969(n)  )-A324337(2*A001969(n)  ) =  A071585(n-1)
A324338(2*A001969(n)+1)-A324337(2*A001969(n)+1) = -A324337(n-1)
A324338(2*A000069(n)  )-A324337(2*A000069(n)  ) = -A071585(n-1)
A324338(2*A000069(n)+1)-A324337(2*A000069(n)+1) =  A324338(n-1) (End)
a(n) = A002487(1+A233279(n)). - Yosu Yurramendi, Dec 27 2019

A324338 a(n) = A002487(1+A006068(n)).

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 3, 2, 1, 4, 5, 3, 4, 3, 2, 5, 1, 5, 7, 4, 7, 5, 3, 8, 5, 4, 3, 7, 2, 7, 8, 5, 1, 6, 9, 5, 10, 7, 4, 11, 9, 7, 5, 12, 3, 11, 13, 8, 6, 5, 4, 9, 3, 10, 11, 7, 2, 9, 12, 7, 11, 8, 5, 13, 1, 7, 11, 6, 13, 9, 5, 14, 13, 10, 7, 17, 4, 15, 18, 11, 11, 9, 7, 16, 5, 17, 19, 12, 3, 14, 19, 11, 18, 13, 8, 21, 7, 6, 5, 11, 4, 13, 14, 9, 3, 13

Offset: 0

Antti Karttunen, Feb 23 2019

nonn

Like in A324337, a few terms preceding each 2^k-th term (here always 1) seem to consist of a batch of nearby Fibonacci numbers (A000045) in some order. For example, a(65533) = 987, a(65534) = 610 and a(65535) = 1597.

For all n > 0 A324338(n)/A324337(n) constitutes an enumeration system of all positive rationals. For all n > 0 A324338(n) + A324337(n) = A071585(n). - Yosu Yurramendi, Oct 22 2019

Antti Karttunen, Table of n, a(n) for n = 0..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65537
Index entries for sequences related to binary expansion of n
Index entries for sequences related to Stern's sequences

Cf. A000045, A002487, A006068, A324288, A324337.

A006068(n)= { my(s=1, ns); while(1, ns = n >> s; if(0==ns, break()); n = bitxor(n, ns); s <<= 1; ); return (n); } \\ From A006068
A002487(n) = { my(s=sign(n), a=1, b=0); n = abs(n); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (s*b); };
A324338(n) = A002487(1+A006068(n));

maxlevel <- 6 # by choice #
b <- 0; A324338 <- 1; A324337 <- 1
for(i in 1:2^maxlevel) {
  b[2*i  ] <-     b[i]
  b[2*i+1] <- 1 - b[i]
  A324338[2*i  ] <- A324338[i]          + A324337[i]*   b[i]
  A324338[2*i+1] <- A324338[i]          + A324337[i]*(1-b[i])
  A324337[2*i  ] <- A324338[i]*(1-b[i]) + A324337[i]
  A324337[2*i+1] <- A324338[i]*   b[i]  + A324337[i]}
#
A324338[1:127]; A324337[1:127]
# Yosu Yurramendi, Oct 22 2019

a(n) = A002487(1+A006068(n)).
a(2^n) = 1 for all n >= 0.
From Yosu Yurramendi, Oct 22 2019: (Start)
a(2^m+2^(m-1)+k) = A324337(2^m+        k), m > 0, 0 <= k < 2^(m-1)
a(2^m+        k) = A324337(2^m+2^(m-1)+k), m > 0, 0 <= k < 2^(m-1). (End)
a(n) = A324337(A063946(n)), n > 0. Yosu Yurramendi, Nov 04 2019
a(n) = A002487(A233279(n)), n > 0. Yosu Yurramendi, Nov 08 2019
From Yosu Yurramendi, Nov 28 2019: (Start)
a(2^(m+1)+k) - a(2^m+k) = A324337(k), m >= 0,  0 <= k < 2^m.
a(A059893(2^(m+1)+A000069(k+1))) - a(A059893(2^m+A000069(k+1))) =  A071585(k), m >= 1,  0 <= k < 2^(m-1).
a(A059893(2^m+ A001969(k+1))) = A071585(k),    m >= 0,  0 <= k < 2^(m-1). (End)
From Yosu Yurramendi, Nov 29 2019: (Start)
For n > 0:
A324338(n) + A324337(n) = A071585(n).
A324338(2*A001969(n)  )-A324337(2*A001969(n)  ) =  A071585(n-1)
A324338(2*A001969(n)+1)-A324337(2*A001969(n)+1) = -A324337(n-1)
A324338(2*A000069(n)  )-A324337(2*A000069(n)  ) = -A071585(n-1)
A324338(2*A000069(n)+1)-A324337(2*A000069(n)+1) =  A324338(n-1) (End)
a(n) = A002487(A233279(n)). Yosu Yurramendi, Dec 27 2019

A339695 Let G be the undirected graph with nodes {g_k, k >= 0} such that for any k >= 0, g_k is connected to g_{k+1} and g_{A006068(k)} is connected to g_{A006068(k+1)}; a(n) is the distance between g_0 and g_n.

Original entry on oeis.org

0, 1, 2, 2, 3, 4, 4, 3, 4, 5, 6, 6, 6, 5, 6, 5, 6, 7, 8, 8, 9, 10, 10, 9, 9, 8, 7, 8, 9, 9, 8, 7, 8, 9, 10, 10, 11, 12, 12, 11, 12, 13, 14, 14, 14, 13, 14, 13, 13, 12, 12, 11, 10, 9, 10, 11, 12, 13, 14, 14, 13, 13, 12, 11, 12, 13, 14, 14, 15, 16, 16, 15, 16

Offset: 0

Rémy Sigrist, Dec 13 2020

nonn

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Dana G. Korssjoen, Biyao Li, Stefan Steinerberger, Raghavendra Tripathi, and Ruimin Zhang, Finding structure in sequences of real numbers via graph theory: a problem list, arXiv:2012.04625, Dec 08, 2020.
Rémy Sigrist, Illustration of initial terms
Rémy Sigrist, PARI program for A339695

See A339731 for a similar sequence.

Cf. A003188, A006068, A339696, A339697.

PARI
```
See Links section.
```

abs(a(n) - a(k)) <= abs(n-k) for any n, k >= 0.

a(n) = A339697(n, 0).

A268719 Triangular table T(n>=0,k=0..n) = A003188(A006068(n) + A006068(k)), read by rows as A(0,0), A(1,0), A(1,1), A(2,0), A(2,1), A(2,2), ...

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Feb 13 2016

			The first fifteen rows of the triangle:
                             0
                           1   3
                         2   6   5
                       3   2   7   6
                     4  12  15  13   9
                   5   4  13  12  11  10
                 6   7   4   5  14  15  12
               7   5  12   4  10  14  13  15
             8  24  27  25  29  31  26  30  17
           9   8  25  24  31  30  27  26  19  18
        10  11   8   9  26  27  24  25  22  23  20
      11   9  24   8  30  26  25  27  18  22  21  23
    12  13  14  15   8   9  10  11  28  29  30  31  24
  13  15  10  14  24   8  11   9  20  28  31  29  25  27
14  10   9  11  27  25   8  24  23  21  28  20  26  30  29

Antti Karttunen, Table of n, a(n) for n = 0..15050; rows 0 .. 172 of the triangular table

Cf. A003188, A006068.
Cf. A002262, A003056, A268715.
Cf. A001477 (left edge), A001969 (right edge).
Cf. A268720 (row sums).

a88[n_] := BitXor[n, Floor[n/2]];
a68[n_] := BitXor @@ Table[Floor[n/2^m], {m, 0, Floor[Log[2, n]]}];
a68[0] = 0;
T[n_, k_] := a88[a68[n] + a68[k]];
Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 19 2019 *)

def a003188(n): return n^(n>>1)
def a006068(n):
    s=1
    while True:
        ns=n>>s
        if ns==0: break
        n=n^ns
        s<<=1
    return n
def T(n, k): return a003188(a006068(n) + a006068(k))
for n in range(21): print([T(n, k) for k in range(n + 1)]) # Indranil Ghosh, Jun 07 2017

(define (A268719 n) (A268715bi (A003056 n) (A002262 n)))

T(n,k) = A003188(A006068(n) + A006068(k)).

a(n) = A268715(A003056(n), A002262(n)). [As a linear sequence.]

A268720 Row sums of A268719: a(n) = Sum_{k=0..n} A003188(A006068(n)+A006068(k)).

Original entry on oeis.org

0, 4, 13, 18, 53, 55, 63, 80, 217, 217, 205, 244, 234, 264, 305, 328, 881, 877, 841, 916, 790, 864, 977, 988, 900, 956, 1021, 1070, 1197, 1235, 1267, 1344, 3553, 3541, 3457, 3604, 3310, 3456, 3681, 3684, 3100, 3244, 3453, 3478, 3917, 3931, 3883, 4048, 3528, 3636, 3757, 3850, 4021, 4111, 4199, 4320, 4745, 4817

Offset: 0

Antti Karttunen, Feb 13 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 0..8191

Row sums of triangle A268719.

Cf. A003188, A006068, A268715.

(define (A268720 n) (add (lambda (k) (A268715bi n k)) 0 n)) ;; Code for A268715bi given in A268715.
(define (add intfun lowlim uplim) (let sumloop ((i lowlim) (res 0)) (cond ((> i uplim) res) (else (sumloop (1+ i) (+ res (intfun i)))))))

a(n) = Sum_{k=0..n} A003188(A006068(n)+A006068(k)).

A268721 Convolution of A006068 (inverse of Gray code) with itself: a(n) = Sum_{k=1..n+1} A006068(k) * A006068(1+n-k).

Original entry on oeis.org

0, 1, 6, 13, 26, 58, 72, 107, 160, 230, 286, 440, 558, 599, 696, 851, 1032, 1298, 1510, 1826, 2122, 2353, 2624, 3294, 3884, 4335, 4870, 5001, 5242, 5722, 6048, 6699, 7424, 8226, 8990, 10166, 11226, 12069, 13048, 14384, 15664, 16885, 18134, 19071, 20094, 21276, 22360, 25150, 27788, 30091, 32582, 34343, 36262

Offset: 0

Antti Karttunen, Feb 13 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 0..1023

Antidiagonal sums of array A268724.

Cf. A006068

(define A268721 (CONVOLVE 1 A006068 A006068)) ;; This version requires Antti Karttunen's IntSeq-library.
;; More stand-alone version:
(define (A268721 n) (add (lambda (k) (* (A006068 k) (A006068 (- (+ n 1) k)))) 1 (+ n 1)))
(define (add intfun lowlim uplim) (let sumloop ((i lowlim) (res 0)) (cond ((> i uplim) res) (else (sumloop (1+ i) (+ res (intfun i)))))))

a(n) = Sum_{k=1..n+1} A006068(k) * A006068(1+n-k).

cp's OEIS Frontend

A268670 a(n) = A006068(A268669(n)).

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A268716 a(n) = 2*A006068(n); main diagonal of A268714.

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A277902 If A010060(n) = 1, a(n) = A000069(A268671(n)), otherwise a(n) = A001969(1+a(A006068(n)/2)).

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A297110 Xor-Moebius transform of A006068, inverse of the binary Gray code.

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A324337 a(n) = A002487(A006068(n)).

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A324338 a(n) = A002487(1+A006068(n)).

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A339695 Let G be the undirected graph with nodes {g_k, k >= 0} such that for any k >= 0, g_k is connected to g_{k+1} and g_{A006068(k)} is connected to g_{A006068(k+1)}; a(n) is the distance between g_0 and g_n.

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A268719 Triangular table T(n>=0,k=0..n) = A003188(A006068(n) + A006068(k)), read by rows as A(0,0), A(1,0), A(1,1), A(2,0), A(2,1), A(2,2), ...

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