A385608 -id:A385608 - cp's OEIS frontend

A385458 Triangle read by rows: T(n,k) = exponent of the highest power of 2 dividing each Fibonomial coefficient fibonomial(n, k).

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

Offset: 0

David Radcliffe, Jun 29 2025

			Triangle begins:
   n\k  0  1  2  3  4  5  6  7  8  9 10 11 12
   0:   0
   1:   0  0
   2:   0  0  0
   3:   0  1  1  0
   4:   0  0  1  0  0
   5:   0  0  0  0  0  0
   6:   0  3  3  2  3  3  0
   7:   0  0  3  2  2  3  0  0
   8:   0  0  0  2  2  2  0  0  0
   9:   0  1  1  0  3  3  0  1  1  0
  10:   0  0  1  0  0  3  0  0  1  0  0
  11:   0  0  0  0  0  0  0  0  0  0  0  0
  12:   0  4  4  3  4  4  1  4  4  3  4  4  0

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.
David Radcliffe, Matrix plot of the first 768 rows
Diana L. Wells, The Fibonacci and Lucas triangles modulo 2, Fibonacci Quart. 32, no. 2 (1994), 111-123. (Theorem 2)

Cf. A000120, A010048, A007814, A065040, A337923, A385456, A385457, A385608.

function T_row(n)
    function T(n, k)
        c(a, b) = 2 * a + b ÷ 6 - count_ones(a)
        (nd, nm) = divrem(n, 3)
        (kd, km) = divrem(k, 3)
        !(nm < km || (kd & (nd - kd)) != 0) && return 0
        c(nd, n) - c(kd, k) - c((n - k) ÷ 3, n - k)
    end
    [T(n, k) for k in 0:n]
end
for n in 0:12 println(T_row(n)) end  # Peter Luschny, Jul 02 2025

A385608[n_] := A385608[n] = 2*# + Quotient[n, 6] - DigitSum[#, 2] & [Quotient[n, 3]];
A385458[n_, k_] := A385608[n] - A385608[k] - A385608[n-k];
Table[A385458[n, k], {n, 0, 15}, {k, 0, n}] (* Paolo Xausa, Jul 04 2025 *)

def b(n): return 2*(n//3) + n//6 - (n//3).bit_count()
def T(n, k): return b(n) - b(k) - b(n-k) # David Radcliffe, Jul 01 2025

T(n, k) = A007814(A010048(n, k)).
T(n, k) = Sum_{i=1..k} (A337923(n+1-i) - A337923(i)).
T(n, k) = b(n) - b(k) - b(n - k), where b(n) = 2*floor(n/3) + floor(n/6) - A000120(floor(n/3)) = A385608(n) is the 2-adic valuation of the product of the first n Fibonacci numbers.
sign(T(n, k)) = 1 - A385456(n, k). - Peter Luschny, Jul 03 2025

A385609 Partial sums of A090740.

Original entry on oeis.org

1, 4, 5, 9, 10, 13, 14, 19, 20, 23, 24, 28, 29, 32, 33, 39, 40, 43, 44, 48, 49, 52, 53, 58, 59, 62, 63, 67, 68, 71, 72, 79, 80, 83, 84, 88, 89, 92, 93, 98, 99, 102, 103, 107, 108, 111, 112, 118, 119, 122, 123, 127, 128, 131, 132, 137, 138, 141, 142, 146, 147, 150, 151

Offset: 1

Paolo Xausa, Jul 04 2025

Prepended with 0, a trisection of A385608.

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A000120, A090740, A385608.

A385609[n_] := 2*n + Quotient[n, 2] - DigitSum[n, 2];
Array[A385609, 100] (* or *)
Accumulate[IntegerExponent[3^Range[100] - 1, 2]]

a(n) = Sum_{k=1..n} A090740(k).
a(n) = 2*n + floor(n/2) - A000120(n).
a(n) = A385608(n*3).

cp's OEIS Frontend

A385458 Triangle read by rows: T(n,k) = exponent of the highest power of 2 dividing each Fibonomial coefficient fibonomial(n, k).

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