A006068 -id:A006068 - cp's OEIS frontend

A268836 Antidiagonal sums of array A268714: a(n) = Sum_{k=0..n} A006068(n)+A006068(n-k).

0, 2, 8, 12, 26, 38, 46, 56, 86, 114, 138, 164, 180, 198, 220, 240, 302, 362, 418, 476, 524, 574, 628, 680, 712, 746, 784, 820, 866, 910, 950, 992, 1118, 1242, 1362, 1484, 1596, 1710, 1828, 1944, 2040, 2138, 2240, 2340, 2450, 2558, 2662, 2768, 2832, 2898, 2968, 3036, 3114, 3190, 3262, 3336, 3430, 3522, 3610, 3700

Offset: 0

Antti Karttunen, Feb 15 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 0..4096

Cf. A006068, A268714.
Cf. also A268837, A268721.
Partial sums of A268716.

(define (A268836 n) (add (lambda (k) (+ (A006068 k) (A006068 (- (+ n 0) k)))) 0 n))
(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} A006068(n)+A006068(n-k).

A268837 Antidiagonal sums of array A268715: a(n) = Sum_{k=0..n} A003188(A006068(n)+A006068(n-k)).

Original entry on oeis.org

0, 2, 7, 18, 17, 48, 56, 80, 67, 122, 136, 194, 204, 268, 281, 328, 291, 378, 396, 498, 510, 640, 675, 792, 790, 886, 965, 1098, 1093, 1208, 1248, 1344, 1227, 1378, 1356, 1530, 1538, 1792, 1815, 2016, 2008, 2218, 2339, 2602, 2619, 2892, 2970, 3208, 3150, 3294, 3385, 3586, 3691, 4012, 4174, 4440, 4367, 4554, 4644

Offset: 0

Antti Karttunen, Feb 15 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 0..4096

Cf. A003188, A006068.

Cf. also A268720, A268836.

(define (A268837 n) (add (lambda (k) (A003188 (+ (A006068 k) (A006068 (- n k))))) 0 n))
(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(n-k)).

A277812 a(n) = the first odious number encountered when starting from k = n and iterating the map k -> A003188(A006068(k)/2).

Original entry on oeis.org

1, 2, 1, 4, 2, 1, 7, 8, 4, 2, 11, 1, 13, 14, 7, 16, 8, 4, 19, 2, 21, 22, 11, 1, 25, 26, 13, 28, 14, 7, 31, 32, 16, 8, 35, 4, 37, 38, 19, 2, 41, 42, 21, 44, 22, 11, 47, 1, 49, 50, 25, 52, 26, 13, 55, 56, 28, 14, 59, 7, 61, 62, 31, 64, 32, 16, 67, 8, 69, 70, 35, 4, 73, 74, 37, 76, 38, 19, 79, 2, 81, 82, 41, 84, 42, 21, 87, 88, 44, 22, 91, 11, 93, 94, 47, 1, 97, 98, 49, 100

Offset: 1

Antti Karttunen, Nov 03 2016

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

Cf. A000069, A001969, A000265, A003188, A006068, A010060, A268669, A277813.
Cf. A277808 (gives the number of such iterations needed to reach a(n) from n).
Cf. A003945 (the positions of 1's in this sequence).

If A010060(n) = 1 [when n is one of the odious numbers, A000069], then a(n) = n, otherwise a(n) = a(A003188(A006068(n)/2)).
Other identities:
a(n) = A000069(A277813(n)).
If A010060(n) = 0 [when n is one of the evil numbers, A001969], then a(n)= a(A000265(n)) [the trailing zeros in binary expansion of n do not affect the result].
For all n >= 1, a(A000069(n)) = A000069(n). [By definition].
For all n > 1, a(A001969(n)) < A001969(n).

A283999 a(n) = A005187(n) XOR A006068(n), where XOR is bitwise-xor (A003987).

Original entry on oeis.org

0, 0, 0, 6, 0, 14, 14, 14, 0, 30, 30, 30, 30, 30, 18, 16, 0, 62, 62, 62, 62, 62, 50, 48, 62, 62, 34, 32, 34, 32, 44, 44, 0, 126, 126, 126, 126, 126, 114, 112, 126, 126, 98, 96, 98, 96, 108, 108, 126, 126, 66, 64, 66, 64, 76, 76, 66, 64, 92, 92, 92, 92, 92, 82, 0, 254, 254, 254, 254, 254, 242, 240, 254, 254, 226, 224, 226, 224, 236, 236, 254, 254, 194, 192, 194

Offset: 0

Antti Karttunen, Mar 20 2017

Antti Karttunen, Table of n, a(n) for n = 0..8192
Index entries for sequences related to binary expansion of n

Cf. A003987, A005187, A006068, A283997.

Table[BitXor[Fold[BitXor, n, Quotient[n, 2^Range[BitLength@ n - 1]]], 2 n - DigitCount[2 n, 2, 1]], {n, 0, 84}] (* Michael De Vlieger, Mar 20 2017, after Jan Mangaldan at A006068 *)

b(n) = if(n<1, 0, b(n\2) + n%2);
A(n) = 2*n - b(2*n);
a(n) = if(n<2, n, 2*a(floor(n/2)) + (n%2 + a(floor(n/2))%2)%2);
for(n=0, 110, print1(bitxor(A(n),a(n)),", ")) \\ Indranil Ghosh, Mar 25 2017

def A(n): return 2*n - bin(2*n)[2:].count("1")
def a(n): return n if n<2 else 2*a(n//2) + (n%2 + a(n//2)%2)%2
print([A(n)^a(n) for n in range(111)]) # Indranil Ghosh, Mar 25 2017

(define (A283999 n) (A003987bi (A005187 n) (A006068 n))) ;; Where A003987bi implements bitwise-XOR (A003987).

a(n) = A005187(n) XOR A006068(n), where XOR is bitwise-xor (A003987).

a(n) = A006068(2*n) XOR A283997(2*n).

A286546 a(n) = A006068(n) - n.

Original entry on oeis.org

0, 0, 1, -1, 3, 1, -2, -2, 7, 5, 2, 2, -4, -4, -3, -5, 15, 13, 10, 10, 4, 4, 5, 3, -8, -8, -7, -9, -5, -7, -10, -10, 31, 29, 26, 26, 20, 20, 21, 19, 8, 8, 9, 7, 11, 9, 6, 6, -16, -16, -15, -17, -13, -15, -18, -18, -9, -11, -14, -14, -20, -20, -19, -21, 63, 61, 58, 58, 52, 52, 53, 51, 40, 40, 41, 39, 43, 41, 38, 38

Offset: 0

Antti Karttunen, May 18 2017

Antti Karttunen, Table of n, a(n) for n = 0..16383
Index entries for sequences related to binary expansion of n

Cf. A006068, A283999, A286547, A286548, A292601.

def A286546(n):
    k, m = n, n>>1
    while m > 0:
        k ^= m
        m >>= 1
    return k-n # Chai Wah Wu, Jun 30 2022

Scheme
```
(define (A286546 n) (- (A006068 n) n))
```

a(n) = A006068(n) - n.

a(n) = -A286548(A006068(n)). - Antti Karttunen, Oct 02 2017

A286547 Restricted growth sequence of A286546 (A006068(n) - n).

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, May 18 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 0..65535

Cf. A006068, A286546, A286549.

rgs_transform(invec) = { my(occurrences = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(occurrences,invec[i]), my(pp = mapget(occurrences, invec[i])); outvec[i] = outvec[pp] , mapput(occurrences,invec[i],i); outvec[i] = u; u++ )); outvec; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
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.
A286546(n) = (A006068(n)-n);
write_to_bfile(0,rgs_transform(vector(16384,n,A286546(n-1))),"b286547.txt");

A339696 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 number of nodes at distance n from g_0.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 5, 3, 6, 7, 6, 5, 8, 9, 10, 5, 8, 10, 16, 12, 15, 12, 12, 7, 14, 18, 18, 16, 20, 17, 18, 7, 14, 18, 18, 17, 25, 29, 33, 21, 28, 30, 31, 22, 25, 24, 22, 11, 18, 24, 38, 31, 39, 35, 37, 25, 42, 46, 37, 29, 37, 33, 30, 11, 18, 24, 38, 31, 39, 35

Offset: 0

Rémy Sigrist, Dec 13 2020

nonn

Rémy Sigrist, Table of n, a(n) for n = 0..4000
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, PARI program for A339696

Cf. A003188, A006068, A339695, A339697.

A339697 Square array T(n, k) read by antidiagonals, n >= 0 and k >= 0; 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)}; T(n, k) is the distance between g_n and g_k.

Original entry on oeis.org

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

Offset: 0

Rémy Sigrist, Dec 13 2020

			Array T(n, k) begins:
  n\k|  0  1  2  3  4  5  6  7  8  9  10  11  12
  ---+------------------------------------------
    0|  0  1  2  2  3  4  4  3  4  5   6   6   6
    1|  1  0  1  1  2  3  3  2  3  4   5   5   5
    2|  2  1  0  1  2  3  2  1  2  3   4   4   4
    3|  2  1  1  0  1  2  2  2  3  4   5   5   5
    4|  3  2  2  1  0  1  1  2  3  4   5   5   4
    5|  4  3  3  2  1  0  1  2  3  4   5   4   3
    6|  4  3  2  2  1  1  0  1  2  3   4   4   4
    7|  3  2  1  2  2  2  1  0  1  2   3   3   3
    8|  4  3  2  3  3  3  2  1  0  1   2   2   2
    9|  5  4  3  4  4  4  3  2  1  0   1   1   2
   10|  6  5  4  5  5  5  4  3  2  1   0   1   2
   11|  6  5  4  5  5  4  4  3  2  1   1   0   1
   12|  6  5  4  5  4  3  4  3  2  2   2   1   0

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, PARI program for A339697
Rémy Sigrist, Colored representation of the table for 0 <= x, y <= 2^10 (where the hue is function of T(x, y))

Cf. A003188, A006068, A339695, A339696.

PARI
```
See Links section.
```

T(n, n) = 0.
T(n, k) = T(k, n).
T(n, k) <= abs(n-k).
T(m, k) <= T(m, n) + T(n, k).
T(n, 0) = A339695(n).

A268722 a(n) = A003188(3*A006068(n)), where A003188 is binary Gray code and A006068 is its inverse.

Original entry on oeis.org

0, 2, 13, 5, 31, 27, 10, 8, 59, 63, 54, 52, 20, 22, 49, 17, 115, 119, 126, 124, 108, 110, 121, 105, 40, 42, 37, 45, 103, 99, 34, 32, 227, 231, 238, 236, 252, 254, 233, 249, 216, 218, 213, 221, 247, 243, 210, 208, 80, 82, 93, 85, 79, 75, 90, 88, 203, 207, 198, 196, 68, 70, 193, 65

Offset: 0

Antti Karttunen, Feb 13 2016

nonn

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

Cf. A003188, A006068.

Row 2 and column 2 of array A268725.

def A268722(n):
    k, m = n, n>>1
    while m > 0:
        k ^= m
        m >>= 1
    return (a:=3*k)^ a>>1 # Chai Wah Wu, Jun 30 2022

(define (A268722 n) (A003188 (* 3 (A006068 n))))

a(n) = A003188(3*A006068(n)).

A268723 Main diagonal of A268725: a(n) = A003188(A006068(n)^2), where A003188 is binary Gray code and A006068 is its inverse.

Original entry on oeis.org

0, 1, 13, 6, 41, 54, 24, 21, 145, 166, 216, 253, 96, 121, 69, 86, 545, 582, 664, 749, 864, 841, 949, 1014, 384, 433, 477, 486, 793, 278, 344, 357, 2113, 2182, 2328, 2509, 2656, 2793, 2901, 2998, 3456, 3537, 3901, 3366, 3641, 3798, 4056, 3973, 1536, 1633, 1709, 1734, 1801, 1910, 1944, 2037, 3313, 3174, 1112, 1053

Offset: 0

Antti Karttunen, Feb 13 2016

nonn

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

Cf. A000290, A003188, A006068.

Main diagonal of array A268725.

def A268723(n):
    k, m = n, n>>1
    while m > 0:
        k ^= m
        m >>= 1
    return (a:=k**2)^ a>>1 # Chai Wah Wu, Jun 30 2022

(define (A268723 n) (A003188 (A000290 (A006068 n))))

a(n) = A003188(A000290(A006068(n))).

cp's OEIS Frontend

A268836 Antidiagonal sums of array A268714: a(n) = Sum_{k=0..n} A006068(n)+A006068(n-k).

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A268837 Antidiagonal sums of array A268715: a(n) = Sum_{k=0..n} A003188(A006068(n)+A006068(n-k)).

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A277812 a(n) = the first odious number encountered when starting from k = n and iterating the map k -> A003188(A006068(k)/2).

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A283999 a(n) = A005187(n) XOR A006068(n), where XOR is bitwise-xor (A003987).

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A286546 a(n) = A006068(n) - n.

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A286547 Restricted growth sequence of A286546 (A006068(n) - n).

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A339696 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 number of nodes at distance n from g_0.

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A339697 Square array T(n, k) read by antidiagonals, n >= 0 and k >= 0; 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)}; T(n, k) is the distance between g_n and g_k.

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A268722 a(n) = A003188(3*A006068(n)), where A003188 is binary Gray code and A006068 is its inverse.

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