A385395 Triangle read by rows: T(n, k) = [A047999(n, k) = 1 or A385456(n, k) = 1], where [.] is the Iverson bracket.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1

Offset

0

Links

Table of n, a(n) for n=0..90.

Michael De Vlieger, Plot of T(n,k), n = 0..511, k = 0..n, at (x,y) = (-n,k-n/2), showing T(n,k) = 1 in black and T(n,k) = 0 in white.

Peter Luschny, Plot of T(n, k) illustrating the overlay structure.

Formula

T(n, k) = [(n & k = k) or ((n mod 3 >= k mod 3) and (~floor(n/3) & floor(k/3) = 0))], where '&' denotes the bitwise 'and', '~' the bitwise complement, and [.] is the Iverson bracket.

Example

Triangle begins:
  [0] 1;
  [1] 1, 1;
  [2] 1, 1, 1;
  [3] 1, 1, 1, 1;
  [4] 1, 1, 0, 1, 1;
  [5] 1, 1, 1, 1, 1, 1;
  [6] 1, 0, 1, 0, 1, 0, 1;
  [7] 1, 1, 1, 1, 1, 1, 1, 1;
  [8] 1, 1, 1, 0, 0, 0, 1, 1, 1;
  [9] 1, 1, 0, 1, 0, 0, 1, 0, 1, 1;

Maple

with(Bits): IB := b -> subs([true=1, false=0], b): # Iverson Bracket
isT := (n, k) -> And(n, k) = k or (modp(n, 3) >= modp(k, 3) and And(Not(iquo(n, 3)), iquo(k, 3)) = 0):
for n from 0 to 12 do seq(IB(isT(n, k)), k = 0..n) od;

Mathematica

isT[n_, k_] := BitAnd[n, k] === k || (Mod[n, 3] >= Mod[k, 3] && BitAnd[BitNot[Quotient[n, 3]], Quotient[k, 3]] === 0);
Table[Boole[isT[n, k]], {n, 0, 12}, {k, 0, n}] // Flatten

Prog

(Python)
def T(n: int, k: int) -> int:
    if n & k == k: return 1
    nd, nm = divmod(n, 3)
    kd, km = divmod(k, 3)
    return int((nm >= km) and ((kd & nd) == kd))
# Or with:
def isT(n, k): return (n & k == k) or ((n % 3 >= k % 3) and (((~(n // 3)) & (k // 3)) == 0))

Crossrefs

Cf. A047999, A385456.

Sequence in context: A108358 A267959 A144384 * A144475 A011758 A015088

Adjacent sequences: A385392 A385393 A385394 * A385396 A385397 A385398

Keyword

nonn,tabl

Author

Peter Luschny, Jul 03 2025

Status

approved