A352909 -id:A352909 - cp's OEIS frontend

A353660 The binary expansions of A352909(n+1, 1) and A352909(n+1, 2) encode respectively the 1's and the 2's in the ternary expansion of a(n).

0, 2, 1, 6, 3, 8, 7, 5, 4, 18, 9, 20, 19, 11, 10, 24, 21, 15, 12, 26, 25, 23, 22, 17, 16, 14, 13, 54, 27, 56, 55, 29, 28, 60, 57, 33, 30, 62, 61, 59, 58, 35, 34, 32, 31, 72, 63, 45, 36, 74, 73, 65, 64, 47, 46, 38, 37, 78, 75, 69, 66, 51, 48, 42, 39, 80, 79, 77

Offset: 0

Rémy Sigrist, May 02 2022

This sequence is a permutation of the nonnegative integers with inverse A353661.

			For n = 42:
- A352909(43, 1) = 9,
- A352909(43, 2) = 2,
- the binary expansion of 9 is "1001",
- the binary expansion of 2 is "10",
- so the ternary expansion of a(42) is "1021",
- and a(42) = 34.

Cf. A005836, A352909, A353661 (inverse), A353662.

b2t(n) = fromdigits(binary(n), 3)
{ n=-1; for (d=0, 2^8-1, for (k=0, d, if (bitand(t1=k, t2=d-k)==0, print1 (b2t(t1) + 2*b2t(t2)", "); if (n++==67, break (2))))) }

a(n) = A005836(A352909(n+1, 1)) + 2*A005836(A352909(n+1, 2)).

a(n) < 3^k iff n < 3^k.

A353662 The binary expansions of A352909(n+1, 1) and A352909(n+1, 2) encode respectively the 1's and the -1's in the balanced ternary expansion of a(n).

Original entry on oeis.org

0, -1, 1, -3, 3, -4, -2, 2, 4, -9, 9, -10, -8, 8, 10, -12, -6, 6, 12, -13, -11, -7, -5, 5, 7, 11, 13, -27, 27, -28, -26, 26, 28, -30, -24, 24, 30, -31, -29, -25, -23, 23, 25, 29, 31, -36, -18, 18, 36, -37, -35, -19, -17, 17, 19, 35, 37, -39, -33, -21, -15, 15

Offset: 0

Rémy Sigrist, May 02 2022

This sequence is a permutation from the nonnegative integers onto the integers (Z).

			For n = 42:
- A352909(43, 1) = 9,
- A352909(43, 2) = 2,
- the binary expansion of 9 is "1001",
- the binary expansion of 2 is "10",
- so the balanced ternary expansion of a(42) is "10T1",
- and a(42) = 25.

Rémy Sigrist, Table of n, a(n) for n = 0..6560

Cf. A005836, A074330, A117966, A352909, A353660.

b2t(n) = fromdigits(binary(n), 3)
{ n=-1; for (d=0, 2^8-1, for (k=0, d, if (bitand(t1=k, t2=d-k)==0, print1 (b2t(t1) - b2t(t2)", "); if (n++==61, break (2))))) }

a(n) = A005836(A352909(n+1, 1)) - A005836(A352909(n+1, 2)).
a(n) = A117966(A353660(n)).
Sum_{k = 0..n} a(k) = 0 iff n = 0 or n belongs to A074330.

A352910 The j-values of pairs (i,j) listed in A352909.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Apr 09 2022

See A295989 for the i-values.

Rémy Sigrist, Table of n, a(n) for n = 1..6563 (values for i+j <= 256)

Cf. A352909, A295989.

A352910list[ij_] := Select[Range[ij, 0, -1], BitAnd[#, ij-#] == 0 &];
Flatten[Array[A352910list, 25, 0]] (* Paolo Xausa, Feb 24 2024 *)

A295989 Irregular triangle T(n, k), read by rows, n >= 0 and 0 <= k < A001316(n): T(n, k) is the (k+1)-th nonnegative number m such that n AND m = m (where AND denotes the bitwise AND operator).

Original entry on oeis.org

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

Offset: 0

Rémy Sigrist, Dec 02 2017

The (n+1)-th row has A001316(n) terms and sums to n * A001316(n) / 2.
For any n >= 0 and k such that 0 <= k < A001316(n):
- if A000120(n) > 0 then T(n, 1) = A006519(n),
- if A000120(n) > 1 then T(n, 2) = 2^A285099(n),
- if A000120(n) > 0 then T(n, A001316(n)/2 - 1) = A053645(n),
- if A000120(n) > 0 then T(n, A001316(n)/2) = 2^A000523(n),
- if A000120(n) > 0 then T(n, A001316(n) - 2) = A129760(n),
- T(n, A001316(n) - 1) = n,
- the six previous relations correspond respectively (when applicable) to the second term, the third term, the pair of central terms, the penultimate term and the last term of a row,
- T(n, k) AND T(n, A001316(n) - k - 1) = 0,
- T(n, k) + T(n, A001316(n) - k - 1) = n,
- T(n, k) = k for any k < A006519(n+1),
- A000120(T(n, k)) = A000120(k).
If we plot (n, T(n,k)) then we obtain a skewed Sierpinski triangle (see Links section).
If interpreted as a flat sequence a(n) for n >= 0:
- a(n) = 0 iff n = A006046(k) for some k >= 0,
- a(n) = 1 iff n = A006046(2*k + 1) + 1 for some k >= 0,
- a(A006046(k) - 1) = k - 1 for any k > 0.

			Triangle begins:
  0:   [0]
  1:   [0, 1]
  2:   [0, 2]
  3:   [0, 1, 2, 3]
  4:   [0, 4]
  5:   [0, 1, 4, 5]
  6:   [0, 2, 4, 6]
  7:   [0, 1, 2, 3, 4, 5, 6, 7]
  8:   [0, 8]
  9:   [0, 1, 8, 9]
  10:  [0, 2, 8, 10]
  11:  [0, 1, 2, 3, 8, 9, 10, 11]
  12:  [0, 4, 8, 12]
  13:  [0, 1, 4, 5, 8, 9, 12, 13]
  14:  [0, 2, 4, 6, 8, 10, 12, 14]
  15:  [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]

Rémy Sigrist, Rows n = 0..256, flattened
Rémy Sigrist, Scatterplot of (n, T(n, k)) for n = 0..1023 and k = 0..A001316(n)-1

Cf. A000120, A000523, A001316 (row lengths), A006046, A006519, A053645, A129760, A285099.

First column of array in A352909.

A295989row[n_] := Select[Range[0, n], BitAnd[#, n-#] == 0 &];
Array[A295989row, 25, 0] (* Paolo Xausa, Feb 24 2024 *)

T(n,k) = if (k==0, 0, n%2==0, 2*T(n\2,k), k%2==0, 2*T(n\2, k\2), 2*T(n\2, k\2)+1)

For any n >= 0 and k such that 0 <= k < A001316(n):
- T(n, 0) = 0,
- T(2*n, k) = 2*T(n, k),
- T(2*n+1, 2*k) = 2*T(n, k),
- T(2*n+1, 2*k+1) = 2*T(n, k) + 1.

A352911 Cantor's List: Pairs (i, j) of relatively prime positive integers sorted first by i + j then by i.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Apr 09 2022

a(2*n-1) / a(2*n) is the n-th fraction in Cantor's enumeration of the positive rational numbers. - Peter Luschny, Oct 10 2023

			The first few pairs are, seen as an irregular triangle:
  [1, 1],
  [1, 2], [2, 1],
  [1, 3], [3, 1],
  [1, 4], [2, 3], [3, 2], [4, 1],
  [1, 5], [5, 1],
  [1, 6], [2, 5], [3, 4], [4, 3], [5, 2], [6, 1],
  [1, 7], [3, 5], [5, 3], [7, 1],
  [1, 8], [2, 7], [4, 5], [5, 4], [7, 2], [8, 1],
  [1, 9], [3, 7], [7, 3], [9, 1],
  ...

N. J. A. Sloane, Table of n, a(n) for n = 1..9914
Georg Cantor, Ein Beitrag zur Mannigfaltigkeitslehre, Journal für die reine und angewandte Mathematik 84 (1878), 242-258, (p. 250).
N. J. A. Sloane, List of the 4957 pairs (i,j) with i+j <= 127. [Note this is not a b-file.]
Index entries for sequences related to enumerating the rationals

Cf. A352909, A020652 or A038566 (i-coordinates), A020653 (j-coordinates), A366191.

CantorsList := proc(upto) local C, F, n, t, count;
C := NULL; count := 0:
for n from 2 while count < upto do
    F := select(t -> igcd(t, n-t) = 1, [$1..n-1]);
    C := C, seq([t, n - t], t = F);
    count := count + nops(F) od:
ListTools:-Flatten([C]) end:
CantorsList(40);  # Peter Luschny, Oct 10 2023

A352911row[n_]:=Select[Array[{#,n-#}&,n-1],CoprimeQ[First[#],Last[#]]&];
Array[A352911row,10,2] (* Generates 10 rows *) (* Paolo Xausa, Oct 10 2023 *)

from math import gcd
from itertools import chain, count, islice
def A352911_gen(): # generator of terms
    return chain.from_iterable((i,n-i) for n in count(2) for i in range(1,n) if gcd(i,n-i)==1)
A352911_list = list(islice(A352911_gen(),30)) # Chai Wah Wu, Oct 10 2023

A054240 Bit-interleaved number addition table; like binary addition but carries shift 2 instead of 1; addition base sqrt(2).

Original entry on oeis.org

0, 1, 1, 2, 4, 2, 3, 3, 3, 3, 4, 6, 8, 6, 4, 5, 5, 9, 9, 5, 5, 6, 16, 6, 12, 6, 16, 6, 7, 7, 7, 7, 7, 7, 7, 7, 8, 18, 12, 18, 16, 18, 12, 18, 8, 9, 9, 13, 13, 17, 17, 13, 13, 9, 9, 10, 12, 10, 24, 18, 20, 18, 24, 10, 12, 10, 11, 11, 11, 11, 19, 19, 19, 19, 11, 11, 11, 11, 12, 14, 32, 14

Offset: 0

Marc LeBrun, Feb 07 2000

			T(3,1)=6 because (0*2 + 1*sqrt(2) + 1*1) + (0*2 + 0*sqrt(2) + 1*1) = (1*2 + 1*sqrt(2) + 0*1) (i.e., base sqrt(2) addition).

Reinhard Zumkeller, Antidiagonals n=0..125 of square array, flattened

Cf. A000695, A054239, A057300, A062880, A352909 (pairs (i,j) such that A(i,j) = i+j).

Cf. A201651 (triangle read by rows).

import Data.Bits (xor, (.&.), shift)
a054240 :: Integer -> Integer -> Integer
a054240 x 0 = x
a054240 x y = a054240 (x `xor` y) (shift (x .&. y) 2)
a054240_adiag n =  map (\k -> a054240 (n - k) k) [0..n]
a054240_square = map a054240_adiag [0..]
-- Reinhard Zumkeller, Dec 03 2011

From Peter Munn, Dec 10 2019: (Start)
A(m,0) = A(0,m) = m.
A(n,k) = A(k,n).
A(n, A(m,k)) = A(A(n,m), k).
A(m,m) = 4*m.
A(2*n, 2*k) = 2*A(n,k).
A(A000695(n), A000695(k)) = A000695(n+k).
A(A000695(n), 2*A000695(k)) = A000695(n) + 2*A000695(k).
A(A000695(n) + 2*A000695(m), k) = A(A000695(n), k) + A(2*A000695(m), k) - k.
A(A057300(n), A057300(k)) = A057300(A(n,k)).
(End)

A353296 Pairs (i,j) of positive integers with at least one common 1-bit sorted first by i+j then by i.

Original entry on oeis.org

1, 1, 1, 3, 2, 2, 3, 1, 2, 3, 3, 2, 1, 5, 3, 3, 5, 1, 1, 7, 2, 6, 3, 5, 4, 4, 5, 3, 6, 2, 7, 1, 2, 7, 3, 6, 4, 5, 5, 4, 6, 3, 7, 2, 1, 9, 3, 7, 4, 6, 5, 5, 6, 4, 7, 3, 9, 1, 4, 7, 5, 6, 6, 5, 7, 4, 1, 11, 2, 10, 3, 9, 5, 7, 6, 6, 7, 5, 9, 3, 10, 2, 11, 1

Offset: 1

Rémy Sigrist and N. J. A. Sloane, Apr 09 2022

Pairs (i,j) with AND(i,j) <> 0.
See A352909 for the other pairs.
There are A048967(n) pairs (i,j) with n = i+j.

			The first pairs are:
    [1, 1],
    [1, 3], [2, 2], [3, 1],
    [2, 3], [3, 2],
    [1, 5], [3, 3], [5, 1],
    [1, 7], [2, 6], [3, 5], [4, 4], [5, 3], [6, 2], [7, 1],
    [2, 7], [3, 6], [4, 5], [5, 4], [6, 3], [7, 2],
    [1, 9], [3, 7], [4, 6], [5, 5], [6, 4], [7, 3], [9, 1],
    ...

Rémy Sigrist, Table of n, a(n) for n = 1..12392 (values for i+j <= 128)

Cf. A004198, A048967, A352909, A353297 (i values), A353298 (j values).

A353296row[ij_] := Select[Array[{#, ij-#} &, ij], BitAnd @@ # > 0 &];
Array[A353296row, 15] (* Paolo Xausa, Feb 24 2024 *)

for (ij=1, 12, for (i=1, ij, j=ij-i; if (bitand(i,j), print1(i", "j", "))))

A362326 Pairs (i, j) of nonnegative integers whose ternary expansions have no common digit 1 sorted first by i + j then by i.

Original entry on oeis.org

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

Offset: 1

Rémy Sigrist, Apr 16 2023

This sequence is to Sierpinski carpet what A352909 is to Sierpinski gasket.
There are A293974(n + 1) pairs (i, j) with n = i + j.
See A362329 for the other pairs.

			The first pairs are:
    (0, 0),
    (0, 1), (1, 0),
    (0, 2), (2, 0),
    (0, 3), (1, 2), (2, 1), (3, 0),
    (0, 4), (1, 3), (2, 2), (3, 1), (4, 0),
    (0, 5), (2, 3), (3, 2), (5, 0),
    (0, 6), (1, 5), (2, 4), (4, 2), (5, 1), (6, 0),
    (0, 7), (1, 6), (2, 5), (5, 2), (6, 1), (7, 0),
    (0, 8), (2, 6), (6, 2), (8, 0),
    ...

Wikipedia, Sierpiński carpet

Cf. A293974, A352909, A362327 (i-values), A362328 (j-values), A362329 (complement).

is(i, j) = { while (i && j, if (i%3==1 && j%3==1, return (0), i\=3; j\=3;);); return (1); }
row(ij) = apply (i -> [i, ij-i], select(i -> is(i, ij-i), [0..ij]))

cp's OEIS Frontend

A353660 The binary expansions of A352909(n+1, 1) and A352909(n+1, 2) encode respectively the 1's and the 2's in the ternary expansion of a(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A353662 The binary expansions of A352909(n+1, 1) and A352909(n+1, 2) encode respectively the 1's and the -1's in the balanced ternary expansion of a(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A352910 The j-values of pairs (i,j) listed in A352909.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A295989 Irregular triangle T(n, k), read by rows, n >= 0 and 0 <= k < A001316(n): T(n, k) is the (k+1)-th nonnegative number m such that n AND m = m (where AND denotes the bitwise AND operator).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A352911 Cantor's List: Pairs (i, j) of relatively prime positive integers sorted first by i + j then by i.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Python

A054240 Bit-interleaved number addition table; like binary addition but carries shift 2 instead of 1; addition base sqrt(2).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Haskell

Formula

A353296 Pairs (i,j) of positive integers with at least one common 1-bit sorted first by i+j then by i.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A362326 Pairs (i, j) of nonnegative integers whose ternary expansions have no common digit 1 sorted first by i + j then by i.

Views

Author