cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A351705 For any nonnegative number n with binary expansion Sum_{k >= 0} b_k * 2^k, a(n) is the numerator of d(n) = Sum_{k >= 0} b_k * 2^A130472(k). See A351706 for the denominators.

Original entry on oeis.org

0, 1, 1, 3, 2, 3, 5, 7, 1, 5, 3, 7, 9, 13, 11, 15, 4, 5, 9, 11, 6, 7, 13, 15, 17, 21, 19, 23, 25, 29, 27, 31, 1, 9, 5, 13, 17, 25, 21, 29, 3, 11, 7, 15, 19, 27, 23, 31, 33, 41, 37, 45, 49, 57, 53, 61, 35, 43, 39, 47, 51, 59, 55, 63, 8, 9, 17, 19, 10, 11, 21
Offset: 0

Views

Author

Rémy Sigrist, Feb 16 2022

Keywords

Comments

The function d is a bijection from the nonnegative integers to the nonnegative dyadic rationals satisfying d(A000695(n)) = n for any n >= 0.

Examples

			For n = 13:
- 13 = 2^0 + 2^2 + 2^3,
- A130472(0) = 0, A130472(2) = 1, A130472(3) = -2,
- d(13) = 2^0 + 2^1 + 2^-2 = 13/4,
- so a(13) = 13.

Links

Crossrefs

Cf. A000120, A000695, A072345, A351706, A351785, A351786.

Programs

  • PARI
    a(n) = { my (d=0, k); while (n, n-=2^k=valuation(n,2); d+=2^((-1)^k*(k+1)\2)); numerator(d) }

Formula

a(A000695(n)) = n.
a(2^k) = A072345(k-1) for any k > 0.
a(2^k-1) = 2^k-1 for any k >= 0.
A000120(a(n)) = A000120(n).

A351785 Symmetric array T(n, k), n, k >= 0, read by antidiagonals; for any m >= 0 with binary expansion Sum_{i >= 0} b_i*2^i, let d(m) = Sum_{i >= 0} b_i * 2^A130472(i); let t be the inverse of d; T(n, k) = t(d(n) + d(k)).

Original entry on oeis.org

0, 1, 1, 2, 4, 2, 3, 3, 3, 3, 4, 6, 1, 6, 4, 5, 5, 4, 4, 5, 5, 6, 16, 6, 5, 6, 16, 6, 7, 7, 7, 7, 7, 7, 7, 7, 8, 18, 5, 18, 16, 18, 5, 18, 8, 9, 9, 16, 16, 17, 17, 16, 16, 9, 9, 10, 12, 10, 17, 18, 20, 18, 17, 10, 12, 10, 11, 11, 11, 11, 19, 19, 19, 19, 11, 11, 11, 11
Offset: 0

Views

Author

Rémy Sigrist, Feb 19 2022

Keywords

Comments

The function d is a bijection from the nonnegative integers to the nonnegative dyadic rationals satisfying d(A000695(n)) = n for any n >= 0.

Examples

			Array T(n, k) begins:
  n\k|   0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15
  ----  --  --  --  --  --  --  --  --  --  --  --  --  --  --  --  --
    0|   0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15
    1|   1   4   3   6   5  16   7  18   9  12  11  14  13  24  15  26
    2|   2   3   1   4   6   7   5  16  10  11   9  12  14  15  13  24
    3|   3   6   4   5   7  18  16  17  11  14  12  13  15  26  24  25
    4|   4   5   6   7  16  17  18  19  12  13  14  15  24  25  26  27
    5|   5  16   7  18  17  20  19  22  13  24  15  26  25  28  27  30
    6|   6   7   5  16  18  19  17  20  14  15  13  24  26  27  25  28
    7|   7  18  16  17  19  22  20  21  15  26  24  25  27  30  28  29
    8|   8   9  10  11  12  13  14  15   2   3   1   4   6   7   5  16
    9|   9  12  11  14  13  24  15  26   3   6   4   5   7  18  16  17
   10|  10  11   9  12  14  15  13  24   1   4   3   6   5  16   7  18

Links

Crossrefs

Cf. A000695, A130472, A351705, A351706, A351786 (multiplication).

Programs

  • PARI
    d(n) = { my (v=0, k); while (n, n-=2^k=valuation(n, 2); v+=2^((-1)^k*(k+1)\2)); v }
t(n) = { my (v=0, k); while (n, n-=2^k=valuation(n, 2); v+=2^if (k>=0, 2*k, -1-2*k)); v }
T(n,k) = t(d(n)+d(k))

Formula

T(A000695(n), A000695(k)) = A000695(n + k).
T(n, k) = T(k, n).
T(m, T(n, k)) = T(T(m, n), k).
T(n, 0) = n.

A351786 Symmetric array T(n, k), n, k >= 0, read by antidiagonals; for any m >= 0 with binary expansion Sum_{i >= 0} b_i*2^i, let d(m) = Sum_{i >= 0} b_i * 2^A130472(i); let t be the inverse of d; T(n, k) = t(d(n) * d(k)).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 8, 3, 0, 0, 4, 10, 10, 4, 0, 0, 5, 1, 12, 1, 5, 0, 0, 6, 3, 5, 5, 3, 6, 0, 0, 7, 9, 18, 16, 18, 9, 7, 0, 0, 8, 11, 15, 20, 20, 15, 11, 8, 0, 0, 9, 32, 25, 17, 65, 17, 25, 32, 9, 0, 0, 10, 34, 40, 21, 23, 23, 21, 40, 34, 10, 0
Offset: 0

Views

Author

Rémy Sigrist, Feb 19 2022

Keywords

Comments

The function d is a bijection from the nonnegative integers to the nonnegative dyadic rationals satisfying d(A000695(n)) = n for any n >= 0.

Examples

			Array T(n, k) begins:
  n\k|  0   1   2   3   4   5    6    7    8    9   10   11   12   13   14   15
  ---+-------------------------------------------------------------------------
    0|  0   0   0   0   0   0    0    0    0    0    0    0    0    0    0    0
    1|  0   1   2   3   4   5    6    7    8    9   10   11   12   13   14   15
    2|  0   2   8  10   1   3    9   11   32   34   40   42   33   35   41   43
    3|  0   3  10  12   5  18   15   25   40   43   33   38   45   58   48   51
    4|  0   4   1   5  16  20   17   21    2    6    3    7   18   22   19   23
    5|  0   5   3  18  20  65   23   70   10   15   12   25   30   75   72   77
    6|  0   6   9  15  17  23   28   74   34   37   43   56   51   96   62  105
    7|  0   7  11  25  21  70   74   88   42   56   38   52   63  109   99  113
    8|  0   8  32  40   2  10   34   42  128  136  160  168  130  138  162  170
    9|  0   9  34  43   6  15   37   56  136  131  170  164  142  144  173  178
   10|  0  10  40  33   3  12   43   38  160  170  130  137  163  172  132  142

Links

Crossrefs

Cf. A000695, A130472, A351705, A351706, A351785 (addition).

Programs

  • PARI
    d(n) = { my (v=0, k); while (n, n-=2^k=valuation(n, 2); v+=2^((-1)^k*(k+1)\2)); v }
t(n) = { my (v=0, k); while (n, n-=2^k=valuation(n, 2); v+=2^if (k>=0, 2*k, -1-2*k)); v }
T(n,k) = t(d(n)*d(k))

Formula

T(A000695(n), A000695(k)) = A000695(n * k).
T(n, k) = T(k, n).
T(m, T(n, k)) = T(T(m, n), k).
T(n, 0) = 0.
T(n, 1) = n.
Showing 1-3 of 3 results.

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