A184617 -id:A184617 - cp's OEIS frontend

A379175 Irregular triangle T(n, k), n >= 0, k = 1..ceiling(2^(A007895(n)-1)); the n-th row lists the nonnegative integers m such that A184617(m) = A003714(n).

0, 1, 2, 4, 3, 5, 8, 7, 9, 6, 10, 16, 15, 17, 14, 18, 12, 20, 11, 13, 19, 21, 32, 31, 33, 30, 34, 28, 36, 27, 29, 35, 37, 24, 40, 23, 25, 39, 41, 22, 26, 38, 42, 64, 63, 65, 62, 66, 60, 68, 59, 61, 67, 69, 56, 72, 55, 57, 71, 73, 54, 58, 70, 74, 48, 80, 47, 49, 79, 81

Offset: 0

Rémy Sigrist, Dec 17 2024

Also the nonnegative terms of A379147, in order of appearance.
This sequence is a permutation of the nonnegative integers with inverse A379176.
This sequence shares graphical features with A368225.

			Triangle T(n, k) begins:
  n   n-th row
  --  --------------
   0  0
   1  1
   2  2
   3  4
   4  3, 5
   5  8
   6  7, 9
   7  6, 10
   8  16
   9  15, 17
  10  14, 18
  11  12, 20
  12  11, 13, 19, 21
  13  32
  14  31, 33
  15  30, 34

Rémy Sigrist, Table of n, a(n) for n = 0..10922 (rows for n = 0..986 flattened)
Index entries for sequences related to Zeckendorf expansion of n
Index entries for sequences that are permutations of the natural numbers

Cf. A003714, A007895, A184617, A368225, A379147, A379176 (inverse).

tozeck(n) = { for (i=0, oo, if (n<=fibonacci(2+i), my (v=0, f); forstep (j=i, 0, -1, if (n>=f=fibonacci(2+j), n-=f; v+=2^j;); if (n==0, return (v););););); }
row(n) = { my (z = tozeck(n), r = [0], b); while (z, z -= b = 2^valuation(z, 2); r = concat([v - b | v <- r], [v + b | v <- r]);); return (select(v -> v >= 0, r)); }

T(n, ceiling(2^(A007895(n)-1))) = A003714(n).

A379147 Irregular triangle T(n, k), n >= 0, k = 1..2^A007895(n), read by rows; the n-th row lists the integers m such that A184617(abs(m)) = A003714(n).

Original entry on oeis.org

0, -1, 1, -2, 2, -4, 4, -5, -3, 3, 5, -8, 8, -9, -7, 7, 9, -10, -6, 6, 10, -16, 16, -17, -15, 15, 17, -18, -14, 14, 18, -20, -12, 12, 20, -21, -19, -13, -11, 11, 13, 19, 21, -32, 32, -33, -31, 31, 33, -34, -30, 30, 34, -36, -28, 28, 36

Offset: 0

Rémy Sigrist, Dec 16 2024

A permutation of the integers (Z).
For any n >= 0:
- in the Zeckendorf expansion of n,
- replace each Fibonacci number, say A000045(2+i) with i >= 0, by 2^i or -2^i,
- the various values obtained make up the n-th row.

			Triangle T(n, k) begins:
  n   n-th row
  --  ----------------------------------
   0  0
   1  -1, 1
   2  -2, 2
   3  -4, 4
   4  -5, -3, 3, 5
   5  -8, 8
   6  -9, -7, 7, 9
   7  -10, -6, 6, 10
   8  -16, 16
   9  -17, -15, 15, 17
  10  -18, -14, 14, 18
  11  -20, -12, 12, 20
  12  -21, -19, -13, -11, 11, 13, 19, 21
  13  -32, 32
  14  -33, -31, 31, 33
  15  -34, -30, 30, 34

Rémy Sigrist, Table of n, a(n) for n = 0..10922 (rows for n = 0..609 flattened)
Index entries for sequences related to Zeckendorf expansion of n

Cf. A000045, A003714, A007895, A184617, A379175.

tozeck(n) = { for (i=0, oo, if (n<=fibonacci(2+i), my (v=0, f); forstep (j=i, 0, -1, if (n>=f=fibonacci(2+j), n-=f; v+=2^j;); if (n==0, return (v););););); }
row(n) = { my (z = tozeck(n), r = [0], b); while (z, z -= b = 2^valuation(z, 2); r = concat([v - b | v <- r], [v + b | v <- r]);); return (r); }

T(n, 1) = -A003714(n).
T(n, 2^A007895(n)) = A003714(n).
T(n, k) = -T(n, 2^A007895(n)+1-k) for k = 1..2^A007895(n).

A178729 a(n) = n XOR 3n, where XOR is bitwise XOR.

Original entry on oeis.org

0, 2, 4, 10, 8, 10, 20, 18, 16, 18, 20, 42, 40, 42, 36, 34, 32, 34, 36, 42, 40, 42, 84, 82, 80, 82, 84, 74, 72, 74, 68, 66, 64, 66, 68, 74, 72, 74, 84, 82, 80, 82, 84, 170, 168, 170, 164, 162, 160, 162, 164, 170, 168, 170, 148, 146, 144, 146, 148, 138, 136, 138, 132, 130

Offset: 0

Dmitry Kamenetsky, Jun 08 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Helena A. Verrill, Note on the negative base xor sequence n⊕-b(-n), arXiv:2506.09509 [math.NT], 2025.

Cf. A048724, A048725, A178731, A178732, A178733, A178734, A178735, A178736. - Robert G. Wilson v, Jun 09 2010

Cf. A184617.

[BitwiseXor(n, 3*n): n in [0..80]]; // Vincenzo Librandi, Jul 11 2017

f[n_] := BitXor[n, 3 n]; Array[f, 70, 0] (* Robert G. Wilson v, Jun 09 2010 *)

a(n) = bitxor(n, 3*n); \\ Michel Marcus, Jul 11 2017

def A178729(n): return n^ 3*n # Chai Wah Wu, Jun 29 2022

a(n) = A005351(n) XOR A005352(n) (conjectured). Proved by Verrill link.

a(n) = 2 * A184617(n). - Alois P. Heinz, Jul 21 2017

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

A184615 Positive parts of the nonadjacent forms for n.

Original entry on oeis.org

0, 1, 2, 4, 4, 5, 8, 8, 8, 9, 10, 16, 16, 17, 16, 16, 16, 17, 18, 20, 20, 21, 32, 32, 32, 33, 34, 32, 32, 33, 32, 32, 32, 33, 34, 36, 36, 37, 40, 40, 40, 41, 42, 64, 64, 65, 64, 64, 64, 65, 66, 68, 68, 69, 64, 64, 64, 65, 66, 64, 64, 65, 64, 64, 64, 65, 66, 68, 68, 69, 72, 72, 72, 73, 74, 80, 80, 81, 80, 80, 80, 81, 82, 84, 84, 85, 128

Offset: 0

Joerg Arndt, Jan 18 2011

nonn

This sequence together with A184616 (negated negative parts) gives the (signed binary) nonadjacent form (NAF) of n, see fxtbook link.
No two adjacent bits in the binary representations of a(n) are 1.
No two adjacent bits in the binary representations of a(n)+A184616(n) are 1.

			The first few nonadjacent forms (NAF) are
(dots are used for zeros for better readability):
     n     binary(n)  NAF(n)
   0:    .......    .......      0 =
   1:    ......1    ......P      1 =  +1
   2:    .....1.    .....P.      2 =  +2
   3:    .....11    ....P.M      3 =  +4 -1
   4:    ....1..    ....P..      4 =  +4
   5:    ....1.1    ....P.P      5 =  +4 +1
   6:    ....11.    ...P.M.      6 =  +8 -2
   7:    ....111    ...P..M      7 =  +8 -1
   8:    ...1...    ...P...      8 =  +8
   9:    ...1..1    ...P..P      9 =  +8 +1
  10:    ...1.1.    ...P.P.     10 =  +8 +2
  11:    ...1.11    ..P.M.M     11 =  +16 -4 -1
  12:    ...11..    ..P.M..     12 =  +16 -4
  13:    ...11.1    ..P.M.P     13 =  +16 -4 +1
  14:    ...111.    ..P..M.     14 =  +16 -2
  15:    ...1111    ..P...M     15 =  +16 -1
  16:    ..1....    ..P....     16 =  +16
  17:    ..1...1    ..P...P     17 =  +16 +1
  18:    ..1..1.    ..P..P.     18 =  +16 +2
This sequence gives the words obtained by keeping the 'P' (sum of positive terms in rightmost column), keeping the 'M' gives A184616 (negative sum of negative terms in rightmost column).

Rémy Sigrist, Table of n, a(n) for n = 0..8192
Pages 61-62 of Matters Computational (The Fxtbook).

A184616 (negated negative parts), A184617 (sums of both parts =A184615+A184616).

A007302 gives the number of nonzero bits ('M' and 'P' in example).

bin2naf[x_] := Module[{xh, x3, c, np, nm},
  xh = BitShiftRight[x, 1];
  x3 = x + xh;
  c = BitXor[xh, x3];
  np = BitAnd[x3, c];
  nm = BitAnd[xh, c];
  Return[{np, nm}]];
a[n_] := bin2naf[n][[1]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, May 30 2019, from PARI *)

bin2naf(x)=
{ /* Compute (nonadjacent) signed binary representation of x: */
    local(xh,x3,c,np,nm);
    xh = x >> 1;
    x3 = x + xh;
    c = bitxor(xh, x3);
    np = bitand(x3, c);  /* bits == +1 */
    nm = bitand(xh, c);  /* bits == -1 */
    return([np,nm]);  /* np-nm==x */
}
{ for(n=0,100, v = bin2naf(n); print1(v[1],", "); ); } /* show terms */
{ for(n=0,100, v = bin2naf(n); print1(v[2],", "); ); } /* terms of A184616 */
{ for(n=0,100, v = bin2naf(n); print1(v[1]+v[2],", "); ); } /* terms of A184617 */
{ for(n=0,100, v = bin2naf(n); print1(v[1]-v[2],", "); ); }  /* == n */

a(n) - A184616(n) = n

a(n) + A184616(n) = A184617(n)

A184616 Negated negative parts of the nonadjacent forms.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 2, 1, 0, 0, 0, 5, 4, 4, 2, 1, 0, 0, 0, 1, 0, 0, 10, 9, 8, 8, 8, 5, 4, 4, 2, 1, 0, 0, 0, 1, 0, 0, 2, 1, 0, 0, 0, 21, 20, 20, 18, 17, 16, 16, 16, 17, 16, 16, 10, 9, 8, 8, 8, 5, 4, 4, 2, 1, 0, 0, 0, 1, 0, 0, 2, 1, 0, 0, 0, 5, 4, 4, 2, 1, 0, 0, 0, 1, 0, 0, 42, 41

Offset: 0

Joerg Arndt, Jan 18 2011

nonn

This sequence together with A184615 (positive parts) gives the (signed binary) nonadjacent form (NAF) of n, see fxtbook link and example in A184615.
No two adjacent bits in the binary representations of a(n) are 1.
No two adjacent bits in the binary representations of a(n)+A184615(n) are 1.

			(see A184615)

Rémy Sigrist, Table of n, a(n) for n = 0..8192
Pages 61-62 of Matters Computational (The Fxtbook).

Cf. A184615 (positive parts), A184617 (sums of both parts =A184615+A184616).

bin2naf[x_] := Module[{xh, x3, c, np, nm},
  xh = BitShiftRight[x, 1];
  x3 = x + xh;
  c = BitXor[xh, x3];
  np = BitAnd[x3, c];
  nm = BitAnd[xh, c];
  Return[{np, nm}]];
a[n_] := bin2naf[n][[2]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, May 30 2019, from PARI code in A184615 *)

PARI
```
(see A184615)
```

A184615(n) - a(n) = n

a(n) + A184615(n) = A184617(n)

A380122 a(n) is the number of integers m (possibly negative) such that the nonzero digits in the nonadjacent form for m appear in the nonadjacent form for n.

Original entry on oeis.org

1, 2, 2, 4, 2, 4, 4, 4, 2, 4, 4, 8, 4, 8, 4, 4, 2, 4, 4, 8, 4, 8, 8, 8, 4, 8, 8, 8, 4, 8, 4, 4, 2, 4, 4, 8, 4, 8, 8, 8, 4, 8, 8, 16, 8, 16, 8, 8, 4, 8, 8, 16, 8, 16, 8, 8, 4, 8, 8, 8, 4, 8, 4, 4, 2, 4, 4, 8, 4, 8, 8, 8, 4, 8, 8, 16, 8, 16, 8, 8, 4, 8, 8, 16, 8

Offset: 0

Rémy Sigrist, Jan 12 2025

			The nonadjacent form for 25 is "10T001" and has 3 nonzero digits, so a(25) = 2^3 = 8.

Rémy Sigrist, Table of n, a(n) for n = 0..8192
Joerg Arndt, Matters Computational (The Fxtbook), pages 61-62.
Wikipedia, Non-adjacent form

Cf. A000120, A001316, A184617, A380123 (corresponding m's).

a(n) = { my (v = 1); while (n, if (n%2, n -= 2 - (n%4); v *= 2; ); n \= 2; ); return (v); }

a(n) = 2^A000120(A184617(n)) = A001316(A184617(n)).

A377477 Consider the nonadjacent form for n, then reverse the digits, leaving any trailing zeros alone, and take the absolute value.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 19, 12, 13, 14, 15, 16, 17, 18, 11, 20, 21, 38, 35, 24, 29, 26, 39, 28, 25, 30, 31, 32, 33, 34, 23, 36, 41, 22, 27, 40, 37, 42, 83, 76, 45, 70, 67, 48, 61, 58, 51, 52, 77, 78, 71, 56, 57, 50, 79, 60, 49, 62, 63, 64, 65, 66, 47

Offset: 0

Rémy Sigrist, Dec 28 2024

A self-inverse permutation of the nonnegative integers.

			For n = 22: the nonadjacent form for 22 is "10T0T0" (where T denotes -1), reversing the digits and leaving any trailing zeros alone yields "T0T010", so a(22) = |- 2^5 - 2^3 + 2^1| =  38.

Rémy Sigrist, Table of n, a(n) for n = 0..10921
Joerg Arndt, Matters Computational (The Fxtbook), pages 61-62.
Rémy Sigrist, PARI program
Wikipedia, Non-adjacent form
Index entries for sequences that are permutations of the natural numbers

See A160652 and A345201 for similar sequences.

Cf. A057889, A184617, A379015.

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

A184617(a(n)) = A057889(A184617(n)).

cp's OEIS Frontend

A379175 Irregular triangle T(n, k), n >= 0, k = 1..ceiling(2^(A007895(n)-1)); the n-th row lists the nonnegative integers m such that A184617(m) = A003714(n).

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A379147 Irregular triangle T(n, k), n >= 0, k = 1..2^A007895(n), read by rows; the n-th row lists the integers m such that A184617(abs(m)) = A003714(n).

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A178729 a(n) = n XOR 3n, where XOR is bitwise XOR.

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A184615 Positive parts of the nonadjacent forms for n.

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A184616 Negated negative parts of the nonadjacent forms.

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A380122 a(n) is the number of integers m (possibly negative) such that the nonzero digits in the nonadjacent form for m appear in the nonadjacent form for n.

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A377477 Consider the nonadjacent form for n, then reverse the digits, leaving any trailing zeros alone, and take the absolute value.

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