Sophia Pi - cp's OEIS frontend

User: Sophia Pi

Sophia Pi has authored 2 sequences.

A367055 Triangle read by rows: T(n, k) = A000120(n) + A000120(k), 0 <= k <= n.

0, 1, 2, 1, 2, 2, 2, 3, 3, 4, 1, 2, 2, 3, 2, 2, 3, 3, 4, 3, 4, 2, 3, 3, 4, 3, 4, 4, 3, 4, 4, 5, 4, 5, 5, 6, 1, 2, 2, 3, 2, 3, 3, 4, 2, 2, 3, 3, 4, 3, 4, 4, 5, 3, 4, 2, 3, 3, 4, 3, 4, 4, 5, 3, 4, 4, 3, 4, 4, 5, 4, 5, 5, 6, 4, 5, 5, 6, 2, 3, 3, 4, 3, 4, 4, 5, 3

Offset: 0

Mithra Karamchedu and Sophia Pi, Nov 03 2023

T(n, k) is the sum of the Hamming weight of n and the Hamming weight of k.

See A365618 for a table read by antidiagonals.

			Triangle begins:
      k=0  1  2  3  4  5
  n=0:  0;
  n=1:  1, 2;
  n=2:  1, 2, 2;
  n=3:  2, 3, 3, 4;
  n=4:  1, 2, 2, 3, 2;
  n=5:  2, 3, 3, 4, 3, 4;
        ...

Cf. A000069, A000120, A000788, A001969, A365618.

T[n_, k_] := DigitCount[n, 2, 1] + DigitCount[k, 2, 1]

from math import comb, isqrt
def A367055(n): return (n-comb(r:=(m:=isqrt(k:=n+1<<1))+(k>m*(m+1)),2)).bit_count()+(r-1).bit_count() # Chai Wah Wu, Nov 11 2024

T(n, k) = A000120(n) + A000120(k).

A365618 Table read by antidiagonals: T(n, k) = A000120(n) + A000120(k).

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 2, 2, 2, 2, 1, 3, 2, 3, 1, 2, 2, 3, 3, 2, 2, 2, 3, 2, 4, 2, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 1, 4, 3, 4, 2, 4, 3, 4, 1, 2, 2, 4, 4, 3, 3, 4, 4, 2, 2, 2, 3, 2, 5, 3, 4, 3, 5, 2, 3, 2, 3, 3, 3, 3, 4, 4, 4, 4, 3, 3, 3, 3, 2, 4, 3, 4, 2, 5, 4, 5, 2

Offset: 0

Mithra Karamchedu and Sophia Pi, Nov 03 2023

T(n, k) is the sum of the Hamming weight of n and the Hamming weight of k.
Picking all points (n, k) such that T(n, k) <= N for some natural number N iteratively generates a Sierpinski-like fractal H. To generate the fractal, fix i and produce the set H_i = {(x, y) in [0, 1)^2 : T(floor(x * 2^i), floor(y * 2^i)) <= i}. Then, define the "limit fractal" H = {(x, y) in [0, 1)^2 : there exists N such that (x, y) is in H_i for all i >= N}.
The table is symmetric, T(n, k) = T(k, n).
See A367055 for a triangle read by rows.

			The table begins:
      k=0 1 2 3 4
  n=0:  0 1 1 2 1 ...
  n=1:  1 2 2 3 2 ...
  n=2:  1 2 2 3 2 ...
  n=3:  2 3 3 4 3 ...
  n=4:  1 2 2 3 2 ...

Mithra Karamchedu, Points (n, k), indicated in black, where T(n, k) <= 12, 0 <= n, k < 2^12.

Cf. A000120, A367055.

T[n_, k_] := DigitCount[n, 2, 1] + DigitCount[k, 2, 1]

T(n, k) = A000120(n) + A000120(k).

If n_1 and n_2 share no 1 bits in common, then T(n_1 + n_2, k) = A000120(n_1) + A000120(n_2) + A000120(k).

cp's OEIS Frontend

User: Sophia Pi

A367055 Triangle read by rows: T(n, k) = A000120(n) + A000120(k), 0 <= k <= n.

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