A334727 -id:A334727 - cp's OEIS frontend

A361832 For any number k >= 0, let T_k be the triangle whose base corresponds to the ternary expansion of k (without leading zeros) and other values, say t above u and v, satisfy t = (-u-v) mod 3; the ternary expansion of a(n) corresponds to the left border of T_n (the most significant digit being at the bottom left corner).

0, 1, 2, 5, 4, 3, 7, 6, 8, 16, 17, 15, 12, 13, 14, 11, 9, 10, 23, 21, 22, 19, 20, 18, 24, 25, 26, 50, 49, 48, 53, 52, 51, 47, 46, 45, 38, 37, 36, 41, 40, 39, 44, 43, 42, 35, 34, 33, 29, 28, 27, 32, 31, 30, 70, 69, 71, 64, 63, 65, 67, 66, 68, 58, 57, 59, 61, 60

Offset: 0

Rémy Sigrist, Mar 26 2023

This sequence is a variant of A334727.

This sequence is a self-inverse permutation of the nonnegative integers that preserves the number of digits and the leading digit in base 3.

			For n = 42: the ternary expansion of 42 is "1120" and the corresponding triangle is as follows:
       2
      2 2
     1 0 1
    1 1 2 0
So the ternary expansion of a(42) is "1122", and a(42) = 44.

Cf. A004488, A048328, A334727, A361818, A361833 (fixed points).

a(n) = { my (d = digits(n, 3), t = vector(#d)); for (k = 1, #d, t[k] = d[1]; d = vector(#d-1, i, (-d[i]-d[i+1]) % 3);); fromdigits(t, 3); }

a(floor(n/3)) = floor(a(n)/3).
a(A004488(n)) = A004488(a(n)).
a(n) = n for any n in A048328.
a(n) = n iff b belongs to A361833.

A335133 Binary interpretation of the left diagonal of the EQ-triangle with first row generated from the binary expansion of n, with most significant bit given by first row.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 10, 13, 12, 14, 15, 16, 17, 18, 19, 22, 23, 20, 21, 26, 27, 24, 25, 28, 29, 30, 31, 32, 33, 35, 34, 36, 37, 39, 38, 44, 45, 47, 46, 40, 41, 43, 42, 53, 52, 54, 55, 49, 48, 50, 51, 57, 56, 58, 59, 61, 60, 62, 63, 64, 65, 66, 67

Offset: 0

Rémy Sigrist, May 24 2020

For any nonnegative number n, the EQ-triangle for n is built by taking as first row the binary expansion of n (without leading zeros), having each entry in the subsequent rows be the EQ of the two values above it (a "1" indicates that these two values are equal, a "0" indicates that these values are different).

This sequence is a self-inverse permutation of the nonnegative numbers.

			For n = 42:
- the binary representation of 42 is "101010",
- the corresponding EQ-triangle is:
         1 0 1 0 1 0
          0 0 0 0 0
           1 1 1 1
            1 1 1
             1 1
              1
- the bits on the left diagonal are: 1, 0, 1, 1, 1, 1,
- so a(42) = 2^5 + 2^3 + 2^2 + 2^1 + 2^0 = 47.

Rémy Sigrist, Table of n, a(n) for n = 0..8192 (n = 0..2^13)
Index entries for sequences related to binary expansion of n
Index entries for sequences that are permutations of the natural numbers

Cf. A055010, A070939, A279645, A334727 (XOR variant).

a(n) = {
    my (b=binary(n), v=0);
    forstep (x=#b-1, 0, -1,
        if (b[1], v+=2^x);
        b=vector(#b-1, k, b[k]==b[k+1])
    );
    return (v)
}

a(floor(n/2)) = floor(a(n)/2).
abs(a(2*n+1) - a(2*n)) = 1.
a(2^k) = 2^k for any k >= 0.
a(2^k+1) = 2^k+1 for any k >= 0.
a(2^k-1) = 2^k-1 for any k >= 0.
Apparently, a(n) + A334727(n) = A055010(A070939(n)) for any n > 0.

A371635 For any number k >= 0, let T_k be the triangle with values in {-1, 0, +1} whose base corresponds to the balanced ternary expansion of k (without leading zeros) and other values, say t above u and v, satisfy t+u+v = 0 mod 3; the balanced ternary expansion of a(n) corresponds to the left border of T_n (the most significant digit being at the bottom left corner).

Original entry on oeis.org

0, 1, 3, 2, 4, 10, 8, 9, 6, 7, 5, 11, 12, 13, 30, 29, 31, 24, 23, 25, 27, 26, 28, 18, 17, 19, 21, 20, 22, 15, 14, 16, 33, 32, 34, 36, 35, 37, 39, 38, 40, 91, 89, 90, 86, 87, 88, 93, 94, 92, 73, 71, 72, 68, 69, 70, 75, 76, 74, 82, 80, 81, 77, 78, 79, 84, 85, 83

Offset: 0

Rémy Sigrist, Mar 30 2024

This sequence is a variant of A334727 and A361832.

This sequence is a self-inverse permutation of the nonnegative integers.

			For n = 42: the balanced ternary expansion of 42 is "1TTT0" (where T denotes -1), and T_42 is as follows:
         T
        0 1
       1 T 0
      0 T T 1
     1 T T T 0
So the balanced ternary expansion of a(42) is "1010T", and a(42) = 89.

Cf. A334727, A361832, A371636.

a(n) = { my (b = [], d); while (n, b = concat(d = Mod(n, 3), b); n = (n-centerlift(d)) / 3;); my (t = vector(#b)); for (i = 1, #t, t[i] = centerlift(b[1]); b = -vector(#b-1, j, b[j]+b[j+1]);); fromdigits(t, 3); }

A334918 Numbers whose XOR-triangles have reflection symmetry.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 14, 15, 16, 17, 21, 22, 24, 27, 30, 31, 32, 33, 40, 45, 51, 54, 62, 63, 64, 65, 72, 73, 85, 86, 93, 94, 96, 99, 104, 107, 118, 119, 126, 127, 128, 129, 153, 158, 165, 168, 182, 189, 195, 200, 214, 219, 224, 231, 254, 255, 256, 257

Offset: 1

Rémy Sigrist, May 16 2020

There are three possible axes of symmetry:
.
.                V
.       U                 W
.          .___._____.
.           \ .     . /
.            \   .   /
.             .     .
.          .   \ . /   .
.       W       \ /       U
.                .
.
.                V
.
- symmetry through axis U-U is only possible for the numbers 0 and 1,
- symmetry through axis V-V corresponds to binary palindromes,
- symmetry through axis W-W corresponds to number k such that A334727(k) is a binary palindrome,
- 0 and 1 are the only terms whose XOR-triangles have the three symmetries,
- XOR-triangles of other terms have only one kind of symmetry.

			The XOR-triangles for a(15) = 21 and a(16) = 22 are as follows (with dots instead of 0's for clarity):
                      1 . 1 . 1      1 . 1 1 .
                       1 1 1 1        1 1 . 1
                        . . .          . 1 1
                         . .            1 .
                          .              1

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Triangles illustrating the initial terms (binary palindromes are rendered in red)
Index entries for sequences related to binary expansion of n
Index entries for sequences related to XOR-triangles

Cf. A006995, A334556 (rotational symmetry), A334727.

is(n) = { my (b=binary(n)); if (b==Vecrev(b), return (1), my (w=#b-1, x=n); for (k=0, w, if (bittest(n,k)!=bittest(x,0), return (0)); x=bitxor(x,x\2)); return (1)) }

A363038 The decimal digits of a(n) correspond to the Gilbreath transform of the decimal digits of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 10, 11, 12, 13, 14, 15, 16, 17, 18, 22, 21, 20, 21, 22, 23, 24, 25, 26, 27, 33, 32, 31, 30, 31, 32, 33, 34, 35, 36, 44, 43, 42, 41, 40, 41, 42, 43, 44, 45, 55, 54, 53, 52, 51, 50, 51, 52, 53, 54, 66, 65, 64, 63, 62, 61, 60, 61

Offset: 0

Rémy Sigrist, May 14 2023

Leading zeros are ignored.

This sequence preserves the number of digits (A055642) as well as the initial digit (A000030).

			For n = 1029: the Gilbreath transform of (1 0 2 9) is (1 1 1 4), so a(1029) = 1114.

Rémy Sigrist, Table of n, a(n) for n = 0..10000
N. J. A. Sloane, Maple code for Gilbreath transform and related arrays

Cf. A000030, A055642, A334727 (base-2 analog).

A363038[n_]:=Module[{d=IntegerDigits[n]},FromDigits[Join[{First[d]},Table[First[d=Abs[Differences[d]]],Length[d]-1]]]];Array[A363038,200,0] (* Paolo Xausa, May 19 2023 *)

a(n, base = 10) = { my (d = digits(n, base), t = vector(#d)); for (i = 1, #d, t[i] = d[1]; d = vector(#d-1, j, abs(d[j+1]-d[j]));); fromdigits(t, base); }

A335049 The prime factorization of a(n) corresponds to the left diagonal of the XOR-triangle built from prime factorization of n, with 2-adic valuation of a(n) given by last row.

Original entry on oeis.org

1, 2, 6, 4, 30, 3, 210, 8, 36, 15, 2310, 24, 30030, 105, 10, 16, 510510, 72, 9699690, 120, 35, 1155, 223092870, 12, 900, 15015, 216, 840, 6469693230, 5, 200560490130, 32, 770, 255255, 21, 9, 7420738134810, 4849845, 5005, 60, 304250263527210, 70

Offset: 1

Rémy Sigrist, May 21 2020

This sequence is a self-inverse permutation of the natural numbers.

This sequence has strong connections with A334727.

			For n = 198:
- 198 = 11^1 * 7^0 * 5^0 * 3^2 * 2^1,
- the corresponding XOR-triangle is:
         1 0 0 2 1
          1 0 2 3
           1 2 1
            3 3
             0
- so a(n) = 11^1 * 7^1 * 5^1 * 3^3 * 2^0 = 10395.

Cf. A006530, A019565, A071178, A334727, A335019.

a(n) = {
    my (f=factor(n),
        m=if (#f~==0, 0, primepi(f[#f~, 1])),
        x=vector(m, k, valuation(n, prime(m+1-k))),
        v=1);
    forstep (i=m, 1, -1,
        v*=prime(i)^x[1];
        x=vector(#x-1, k, bitxor(x[k], x[k+1]));
    );
    v
}

a(n) = n iff n is a power of 2.
a(n^2) = a(n)^2.
a(A019565(n)) = A019565(A334727(n)).
A006530(a(n)) = A006530(n).
A071178(a(n)) = A071178(n).

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A335133 Binary interpretation of the left diagonal of the EQ-triangle with first row generated from the binary expansion of n, with most significant bit given by first row.

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A334918 Numbers whose XOR-triangles have reflection symmetry.

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A363038 The decimal digits of a(n) correspond to the Gilbreath transform of the decimal digits of n.

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A335049 The prime factorization of a(n) corresponds to the left diagonal of the XOR-triangle built from prime factorization of n, with 2-adic valuation of a(n) given by last row.

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