A385456 Triangle read by rows, formed by reading Fibonomial coefficients (A010048) mod 2.

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

Offset

0

Links

Paolo Xausa, Table of n, a(n) for n = 0..11475 (rows 0..150 of triangle, flattened).

Donald E. Knuth and Herbert S. Wilf, The power of a prime that divides a generalized binomial coefficient, J. Reine Angew. Math., 396:212-219, 1989.

Romeo Meštrović, Lucas' theorem: its generalizations, extensions and applications (1878--2014), arXiv preprint arXiv:1409.3820 [math.NT], 2014.

Diana L. Wells, The Fibonacci and Lucas triangles modulo 2, Fibonacci Quart. 32, no. 2 (1994), 111-123. (Theorem 2)

Formula

a(n) = A010048(n) mod 2.

T(n, k) = binomial([n/3], [k/3]) mod 2 if (n mod 3) >= (k mod 3), 0 otherwise.

Example

Triangle begins:
  n\k: 0  1  2  3  4  5  6  7  8  9 10 11
   0:  1
   1:  1  1
   2:  1  1  1
   3:  1  0  0  1
   4:  1  1  0  1  1
   5:  1  1  1  1  1  1
   6:  1  0  0  0  0  0  1
   7:  1  1  0  0  0  0  1  1
   8:  1  1  1  0  0  0  1  1  1
   9:  1  0  0  1  0  0  1  0  0  1
  10:  1  1  0  1  1  0  1  1  0  1  1
  11:  1  1  1  1  1  1  1  1  1  1  1  1

Mathematica

A385456[n_, k_] := Boole[Mod[n, 3] >= Mod[k, 3] && (BitAnd[#, Quotient[n, 3] - #] & [Quotient[k, 3]]) == 0];
Table[A385456[n, k], {n, 0, 15}, {k, 0, n}] (* Paolo Xausa, Jul 04 2025, after Peter Luschny *)

Prog

(Julia)
function A385456row(n)
    function T(n, k)
        (nd, nm) = divrem(n, 3)
        (kd, km) = divrem(k, 3)
        return nm < km || (kd & (nd - kd)) != 0 ? 0 : 1
    end
    [T(n, k) for k in 0:n]
end  # Peter Luschny, Jul 02 2025
(Python)
def A385456row(n: int) -> list[int]:
    nd, nm = divmod(n, 3)
    return [1 if nm >= km and (kd & (nd - kd)) == 0 else 0 for k in range(n + 1) for kd, km in [divmod(k, 3)]]
for n in range(13): print(A385456row(n))  # Peter Luschny, Jul 03 2025

Crossrefs

Cf. A010048, A047999, A385457 (row sums), A385458.

Sequence in context: A185917 A143104 A127236 * A117947 A385734 A175860

Adjacent sequences: A385453 A385454 A385455 * A385457 A385458 A385459

Keyword

nonn,tabl

Author

David Radcliffe, Jun 29 2025

Status

approved