A001196 -id:A001196 - cp's OEIS frontend

A224915 a(n) = Sum_{k=0..n} n XOR k where XOR is the bitwise logical exclusive-or operator.

0, 1, 5, 6, 22, 23, 27, 28, 92, 93, 97, 98, 114, 115, 119, 120, 376, 377, 381, 382, 398, 399, 403, 404, 468, 469, 473, 474, 490, 491, 495, 496, 1520, 1521, 1525, 1526, 1542, 1543, 1547, 1548, 1612, 1613, 1617, 1618, 1634, 1635, 1639, 1640, 1896, 1897, 1901, 1902, 1918

Offset: 0

Alex Ratushnyak, Apr 19 2013

			a(2) = (0 xor 2) + (1 xor 2) = 2 + 3 = 5.

Michael De Vlieger, Table of n, a(n) for n = 0..10000

Cf. A001196 (bit doubling).
Row sums of A051933.
Other sums: A222423 (AND), A350093 (OR), A265736 (IMPL), A350094 (CNIMPL), A004125 (mod).

read("transforms"):
A051933 := proc(n,k)
    XORnos(n,k) ;
end proc:
A224915 := proc(n)
    add(A051933(n,k),k=0..n) ;
end proc: # R. J. Mathar, Apr 26 2013
# second Maple program:
with(MmaTranslator[Mma]):
seq(add(BitXor(n,i),i=0..n),n=0..60); # Ridouane Oudra, Dec 09 2020

Array[Sum[BitXor[#, k], {k, 0, #}] &, 53, 0] (* Michael De Vlieger, Dec 09 2020 *)

a(n) = sum(k=0, n, bitxor(n, k)); \\ Michel Marcus, Jun 08 2019

a(n) = (3*fromdigits(binary(n),4) - n) >>1; \\ Kevin Ryde, Dec 17 2021

for n in range(59):
    s = 0
    for k in range(n):  s += n ^ k
    print(s, end=',')

def A224915(n): return 3*int(bin(n)[2:],4)-n>>1 # Chai Wah Wu, Aug 21 2023

a(n) = Sum_{j=1..n} 4^(v_2(j)), where v_2(j) is the exponent of highest power of 2 dividing j. - Ridouane Oudra, Jun 08 2019
a(n) = n + 3*Sum_{j=1..floor(log_2(n))} 4^(j-1)*floor(n/2^j), for n>=1. - Ridouane Oudra, Dec 09 2020
From Kevin Ryde, Dec 17 2021: (Start)
a(2*n+b) = 4*a(n) + n + b where b = 0 or 1.
a(n) = (A001196(n) - n)/2.
a(n) = A350093(n) - A222423(n), being XOR = OR - AND.
(End)

A338086 Duplicate the ternary digits of n, so each 0, 1 or 2 becomes 00, 11 or 22 respectively.

Original entry on oeis.org

0, 4, 8, 36, 40, 44, 72, 76, 80, 324, 328, 332, 360, 364, 368, 396, 400, 404, 648, 652, 656, 684, 688, 692, 720, 724, 728, 2916, 2920, 2924, 2952, 2956, 2960, 2988, 2992, 2996, 3240, 3244, 3248, 3276, 3280, 3284, 3312, 3316, 3320, 3564, 3568, 3572, 3600, 3604

Offset: 0

Kevin Ryde, Oct 09 2020

Also, numbers whose ternary digit runs are all even lengths (including 0 reckoned as no digits at all).  Also, change ternary digits 0,1,2 to base 9 digits 0,4,8, and hence numbers which can be written in base 9 using only digits 0,4,8.
Digit duplication 00,11,22 can be compared to A037314 which is 0 above each so 00,01,02, or A208665 which is 0 below each so 00,10,20.  Duplication is the sum of these, or any one is a suitable multiple of another (*3, *4, etc).
This sequence is the points on the X=Y diagonal of the ternary Z-order curve (see example table in A163328).  The Z-order curve takes a point number p and splits its ternary digits alternately to X and Y coordinates so X(p) = A163325(p) and Y(p) = A163326(p).  Duplicate digits in a(n) are the diagonal X(a(n)) = Y(a(n)) = n.

			n=73 is ternary 2201 which duplicates to 22220011 ternary = 8804 base 9 = 6484 decimal.

Kevin Ryde, Table of n, a(n) for n = 0..6560

Cf. A037314, A208665, A163325, A163326, A163328.
Cf. A020331 (ternary concatenation).
Digit duplication in other bases: A001196, A338754.

PARI
```
a(n) = fromdigits(digits(n,3),9)<<2;
```

from gmpy2 import digits
def A338086(n): return int(''.join(d*2 for d in digits(n,3)),3) # Chai Wah Wu, May 07 2022

a(n) = A037314(n) + A208665(n) = 4*A037314(n) = (4/3)*A208665(n).

a(n) = 4*Sum_{i=0..k} d[i]*9^i where the ternary expansion of n is n = Sum_{i=0..k} d[i]*3^i with digits d[i]=0,1,2.

A097252 Numbers whose set of base 6 digits is {0,5}.

Original entry on oeis.org

0, 5, 30, 35, 180, 185, 210, 215, 1080, 1085, 1110, 1115, 1260, 1265, 1290, 1295, 6480, 6485, 6510, 6515, 6660, 6665, 6690, 6695, 7560, 7565, 7590, 7595, 7740, 7745, 7770, 7775, 38880, 38885, 38910, 38915, 39060, 39065, 39090, 39095, 39960, 39965

Offset: 0

Ray Chandler, Aug 03 2004

n such that there exists a permutation p_1, ..., p_n of 1, ..., n such that i + p_i is a power of 6 for every i.

Vincenzo Librandi, Table of n, a(n) for n = 0..500

Cf. A001196, A005823, A007088, A033043, A097251-A097262.

[n: n in [0..40000] | Set(IntegerToSequence(n, 6)) subset {0, 5}]; // Vincenzo Librandi, May 25 2012

fQ[n_]:=Union@Join[{0,5},IntegerDigits[n,6]]=={0,5};Select[Range[0,40000],fQ] (* Vincenzo Librandi, May 25 2012 *)
FromDigits[#,6]&/@Tuples[{ 0,5},6] (* Harvey P. Dale, Aug 15 2021 *)

def A079252(n): return 5*int(bin(n)[2:],6) # Chai Wah Wu, Apr 04 2025

a(n) = 5*A033043(n).

a(2n) = 6*a(n), a(2n+1) = a(2n)+5.

A097254 Numbers whose set of base 8 digits is {0,7}.

Original entry on oeis.org

0, 7, 56, 63, 448, 455, 504, 511, 3584, 3591, 3640, 3647, 4032, 4039, 4088, 4095, 28672, 28679, 28728, 28735, 29120, 29127, 29176, 29183, 32256, 32263, 32312, 32319, 32704, 32711, 32760, 32767, 229376, 229383, 229432, 229439, 229824

Offset: 1

Ray Chandler, Aug 03 2004

n such that there exists a permutation p_1, ..., p_n of 1, ..., n such that i + p_i is a power of 8 for every i.

Vincenzo Librandi, Table of n, a(n) for n = 1..100

Cf. A001196, A005823, A097251-A097262.

[n: n in [0..250000] | Set(IntegerToSequence(n, 8)) subset {0, 7}]; // Vincenzo Librandi, May 25 2012

fQ[n_]:=Union@Join[{0,7},IntegerDigits[n,8]]=={0,7};Select[Range[0,300000],fQ] (* Vincenzo Librandi, May 25 2012 *)
FromDigits[#,8]&/@Tuples[{0,7},6] (* Harvey P. Dale, Aug 10 2021 *)

a[1]:0$ a[n]:=8*a[floor((n+1)/2)]+7*(1+(-1)^n)/2$ makelist(a[n], n, 1, 37); /* Bruno Berselli, May 25 2012 */

a(n) = 7*fromdigits(binary(n-1), 8) \\ Rémy Sigrist, Dec 06 2018

a(n) = 7*A033045(n-1).

a(2n-1) = 8*a(n), a(2n) = 8*a(n)+7.

A004468 a(n) = Nim product 3 * n.

Original entry on oeis.org

0, 3, 1, 2, 12, 15, 13, 14, 4, 7, 5, 6, 8, 11, 9, 10, 48, 51, 49, 50, 60, 63, 61, 62, 52, 55, 53, 54, 56, 59, 57, 58, 16, 19, 17, 18, 28, 31, 29, 30, 20, 23, 21, 22, 24, 27, 25, 26, 32, 35, 33, 34, 44, 47, 45, 46, 36, 39, 37, 38, 40, 43, 41, 42, 192, 195, 193, 194, 204, 207, 205

Offset: 0

N. J. A. Sloane

From Jianing Song, Aug 10 2022: (Start)
Write n in quaternary (base 4), then replace each 1,2,3 by 3,1,2.
This is a permutation of the natural numbers; A006015 is the inverse permutation (since the nim product of 2 and 3 is 1). (End)

J. H. Conway, On Numbers and Games. Academic Press, NY, 1976, pp. 51-53.

Alois P. Heinz, Table of n, a(n) for n = 0..16383 (first 1001 terms from R. J. Mathar)
Index entries for sequences related to Nim-multiplication
Index entries for sequences that are permutations of the natural numbers

Row 3 of array in A051775.

Cf. A006015, A004469-A004480, A000695, A062880, A001196.

read("transforms") ;
# insert Maple procedures nimprodP2() and A051775() of the b-file in A051775 here.
A004468 := proc(n)
        A051775(3,n) ;
end proc:
L := [seq(A004468(n),n=0..1000)] ; # R. J. Mathar, May 28 2011
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 0,
      a(iquo(n, 4, 'r'))*4+[0, 3, 1, 2][r+1])
    end:
seq(a(n), n=0..70);  # Alois P. Heinz, Jan 25 2022

a[n_] := a[n] = If[n == 0, 0, {q, r} = QuotientRemainder[n, 4]; a[q]*4 + {0, 3, 1, 2}[[r + 1]]];
Table[a[n], {n, 0, 70}] (* Jean-François Alcover, May 20 2022, after Alois P. Heinz *)

a(n) = my(v=digits(n, 4), w=[0,3,1,2]); for(i=1, #v, v[i] = w[v[i]+1]); fromdigits(v, 4) \\ Jianing Song, Aug 10 2022

def a(n, D=[0, 3, 1, 2]):
    r, k = 0, 0
    while n>0: r+=D[n%4]*4**k; n//=4; k+=1
    return r
# Onur Ozkan, Mar 07 2023

a(n) = A051775(3,n).
From Jianing Song, Aug 10 2022: (Start)
a(n) = 3*n if n has only digits 0 or 1 in quaternary (n is in A000695). Otherwise, a(n) < 3*n.
a(n) = n/2 if n has only digits 0 or 2 in quaternary (n is in A062880). Otherwise, a(n) > n/2.
a(n) = 2*n/3 if and only if n has only digits 0 or 3 in quaternary (n is in A001196). Proof: let n = Sum_i d_i*4^i, d(i) = 0,1,2,3. Write A = Sum_{d_i=1} 4^i, B = Sum_{d_i=2} 4^i, then a(n) = 2*n/3 if and only if 3*A + B = 2/3*(A + 2*B), or B = 7*A. If A != 0, then A is of the form (4*s+1)*4^t, but 7*A is not of this form. So the only possible case is A = B = 0, namely n has only digits 0 or 3. (End)

More terms from Erich Friedman

A006015 Nim product 2*n.

Original entry on oeis.org

0, 2, 3, 1, 8, 10, 11, 9, 12, 14, 15, 13, 4, 6, 7, 5, 32, 34, 35, 33, 40, 42, 43, 41, 44, 46, 47, 45, 36, 38, 39, 37, 48, 50, 51, 49, 56, 58, 59, 57, 60, 62, 63, 61, 52, 54, 55, 53, 16, 18, 19, 17, 24, 26, 27, 25, 28, 30, 31, 29, 20, 22, 23, 21, 128, 130, 131, 129, 136, 138, 139

Offset: 0

N. J. A. Sloane

From Jianing Song, Aug 10 2022: (Start)
Write n in quaternary (base 4), then replace each 1,2,3 by 2,3,1.
This is a permutation of the natural numbers; A004468 is the inverse permutation (since the nim product of 2 and 3 is 1). (End)

J. H. Conway, On Numbers and Games. Academic Press, NY, 1976, pp. 51-53.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..16383 (first 1001 terms from R. J. Mathar)
Index entries for sequences related to Nim-multiplication
Index entries for sequences that are permutations of the natural numbers

Row 2 of array in A051775.

Cf. A004468-A004480, A000695, A062880, A001196.

a:= proc(n) option remember; `if`(n=0, 0,
      a(iquo(n, 4, 'r'))*4+[0, 2, 3, 1][r+1])
    end:
seq(a(n), n=0..70);  # Alois P. Heinz, Jan 25 2022

a[n_] := a[n] = If[n == 0, 0, {q, r} = QuotientRemainder[n, 4]; a[q]*4 + {0, 2, 3, 1}[[r + 1]]];
Table[a[n], {n, 0, 70}] (* Jean-François Alcover, May 20 2022, after Alois P. Heinz *)

a(n) = my(v=digits(n, 4), w=[0,2,3,1]); for(i=1, #v, v[i] = w[v[i]+1]); fromdigits(v, 4) \\ Jianing Song, Aug 10 2022

def a(n, D=[0, 2, 3, 1]):
    r, k = 0, 0
    while n>0: r+=D[n%4]*4**k; n//=4; k+=1
    return r
# Onur Ozkan, Mar 07 2023

From Jianing Song, Aug 10 2022: (Start)
a(n) = A051775(2,n).
a(n) = 2*n if n has only digits 0 or 1 in quaternary (n is in A000695). Otherwise, a(n) < 2*n.
a(n) = n/3 if n has only digits 0 or 3 in quaternary (n is in A001196). Otherwise, a(n) > n/3.
a(n) = 3*n/2 if and only if n has only digits 0 or 2 in quaternary (n is in A062880). Proof: let n = Sum_i d_i*4^i, d(i) = 0,1,2,3. Write A = Sum_{d_i=1} 4^i, B = Sum_{d_i=3} 4^i, then a(n) = 3*n/2 if and only if 2*A + B = 3/2*(A + 3*B), or A = 7*B. If B != 0, then B is of the form (4*s+1)*4^t, but 7*B is not of this form. So the only possible case is A = B = 0, namely n has only digits 0 or 2. (End)

More terms from Erich Friedman.

A350093 a(n) = Sum_{k=0..n} n OR k where OR is the bitwise logical OR operator (A003986).

Original entry on oeis.org

0, 2, 7, 12, 26, 34, 45, 56, 100, 114, 131, 148, 174, 194, 217, 240, 392, 418, 447, 476, 514, 546, 581, 616, 684, 722, 763, 804, 854, 898, 945, 992, 1552, 1602, 1655, 1708, 1770, 1826, 1885, 1944, 2036, 2098, 2163, 2228, 2302, 2370, 2441, 2512, 2712, 2786, 2863

Offset: 0

Kevin Ryde, Dec 14 2021

The effect of n OR k is to force a 1-bit at all bit positions where n has a 1-bit, which means n*(n+1) in the sum. Bits of k where n has a 0-bit are NOT(n) AND k = n CNIMPL k so that a(n) = A350094(n) + n*(n+1).

Winston de Greef, Table of n, a(n) for n = 0..10000

Cf. A003986 (bitwise OR), A001196 (bit doubling).
Row sums of A080098.
Other sums: A222423 (AND), A224915 (XOR), A265736 (IMPL), A350094 (CNIMPL).

a(n) = (3*(n^2 + fromdigits(binary(n),4)) + 2*n) >> 2;

a(n) = ((3*n+2)*n + A001196(n)) / 4.
a(2*n) = 4*a(n) - n.
a(2*n+1) = 4*a(n) + 2*n + 2.
a(n) = A222423(n) + A224915(n), being OR = AND + XOR.

A371442 For any positive integer n with binary digits (b_1, ..., b_w) (where b_1 = 1), the binary digits of a(n) are (b_1, b_3, ..., b_{2*ceiling(w/2)-1}); a(0) = 0.

Original entry on oeis.org

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

Offset: 0

Rémy Sigrist, Mar 24 2024

In other words, we keep odd-indexed bits.

For any v > 0, the value v appears A003945(A070939(v)) times in the sequence.

			The first terms, in decimal and in binary, are:
  n   a(n)  bin(n)  bin(a(n))
  --  ----  ------  ---------
   0     0       0          0
   1     1       1          1
   2     1      10          1
   3     1      11          1
   4     2     100         10
   5     3     101         11
   6     2     110         10
   7     3     111         11
   8     2    1000         10
   9     2    1001         10
  10     3    1010         11
  11     3    1011         11
  12     2    1100         10
  13     2    1101         10
  14     3    1110         11
  15     3    1111         11

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

See A371459 for the sequence related to even-indexed bits.
See A059905 and A063694 for similar sequences.
Cf. A000695, A001196, A003945, A070939, A165199, A371461.

A371442[n_] := FromDigits[IntegerDigits[n, 2][[1;;-1;;2]], 2];
Array[A371442, 100, 0] (* Paolo Xausa, Mar 28 2024 *)

a(n) = { my (b = binary(n)); fromdigits(vector(ceil(#b/2), k, b[2*k-1]), 2); }

Python
```
def a(n): return int(bin(n)[::2], 2)
```

a(A000695(n)) = n.
a(A001196(n)) = n.
a(A165199(n)) = a(n).

A097261 Numbers whose set of base 15 digits is {0,E}, where E base 15 = 14 base 10.

Original entry on oeis.org

0, 14, 210, 224, 3150, 3164, 3360, 3374, 47250, 47264, 47460, 47474, 50400, 50414, 50610, 50624, 708750, 708764, 708960, 708974, 711900, 711914, 712110, 712124, 756000, 756014, 756210, 756224, 759150, 759164, 759360, 759374, 10631250

Offset: 0

Ray Chandler, Aug 03 2004

n such that there exists a permutation p_1, ..., p_n of 1, ..., n such that i + p_i is a power of 15 for every i.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A001196, A005823, A097251-A097262.

[n: n in [0..4500000] | Set(IntegerToSequence(n, 15)) subset {0, 14}]; // Vincenzo Librandi, Jun 05 2012

f[n_] := FromDigits[ IntegerDigits[n, 2] /. {1 -> 14}, 15]; Array[f, 33, 0] (* or *)
FromDigits[#, 15] & /@ Tuples[{0, 14}, 6] (* Harvey P. Dale, Sep 22 2011 *) (* or much slower *)
fQ[n_] := Union@ Join[{0, 14}, IntegerDigits[n, 15]] == {0, 14}; Select[ Range[0, 10634414 ], fQ] (* Robert G. Wilson v, May 12 2012 *)

a(n) = 14*A033051(n).

a(2n) = 15*a(n), a(2n+1) = a(2n)+14.

A332205 a(n) is the imaginary part of f(n) defined by f(0) = 0, and f(n+1) = f(n) + g((1+i)^(A065359(n) mod 8)) (where g(z) = z/gcd(Re(z), Im(z)) and i denotes the imaginary unit).

Original entry on oeis.org

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

Offset: 0

Rémy Sigrist, Feb 07 2020

Looks much like A005536, in particular in respect of its symmetries of scale (compare the scatterplots). - Peter Munn, Jun 21 2021

Rémy Sigrist, Table of n, a(n) for n = 0..16384
Larry Riddle, Koch Curve
Rémy Sigrist, PARI program for A332205
Index entries for sequences related to coordinates of 2D curves

Cf. A005536, A007052, A065359, A332204 (real part and additional comments), A332206 (positions of 0's, cf. A001196).

A065359[0] = 0;
A065359[n_] := -Total[(-1)^PositionIndex[Reverse[IntegerDigits[n, 2]]][1]];
g[z_] := z/GCD[Re[z], Im[z]];
Module[{n = 0}, Im[NestList[# + g[(1+I)^A065359[n++]] &, 0, 100]]] (* Paolo Xausa, Aug 28 2024 *)

PARI
```
\\ See Links section.
```

a(2^(2*k-1)) = A007052(k) for any k >= 0.

a(4^k-m) = a(m) for any k >= 0 and m = 0..4^k.

cp's OEIS Frontend

A224915 a(n) = Sum_{k=0..n} n XOR k where XOR is the bitwise logical exclusive-or operator.

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A338086 Duplicate the ternary digits of n, so each 0, 1 or 2 becomes 00, 11 or 22 respectively.

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A097252 Numbers whose set of base 6 digits is {0,5}.

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A097254 Numbers whose set of base 8 digits is {0,7}.

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A004468 a(n) = Nim product 3 * n.

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A006015 Nim product 2*n.

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