A327991 The complementary Fibonacci codes. Irregular triangle T(n, k) with n >= 0 and 0 <= k < A000045(n+1).

1, 2, 2, 6, 2, 6, 30, 2, 6, 30, 10, 210, 2, 6, 30, 10, 210, 42, 70, 2310, 2, 6, 30, 10, 210, 42, 70, 2310, 14, 330, 462, 770, 30030, 2, 6, 30, 10, 210, 42, 70, 2310, 14, 330, 462, 770, 30030, 66, 110, 2730, 154, 4290, 6006, 10010, 510510

Offset: 0

Author

Peter Luschny, Oct 09 2019

Keywords

• nonn
• tabf

Comments

The complementary Fibonacci codes are binary strings enumerated in an irregular triangle CF(n, k). The first few are shown below in the Example section. The complementary Fibonacci codes are the bitwise complements of the Fibonacci codes described in A327990, in ascending order.

The complementary Fibonacci codes are represented here through

T(n, k) = Product_{j=0..m} p(j)^c(j),

where p(j) is the j-th prime number, c = CF(n, k) and m = length(CF(n, k)).

Examples

			The complementary Fibonacci codes start:
[0] [[]]
[1] [[1]]
[2] [[1][11]]
[3] [[1][11][111]]
[4] [[1][11][111][101][1111]]
[5] [[1][11][111][101][1111][1101][1011][11111]]
[6] [[1][11][111][101][1111][1101][1011][11111][1001][11101][11011][10111][111111]]
[7] [[1][11][111][101][1111][1101][1011][11111][1001][11101][11011][10111][111111] [11001][10101][111101][10011][111011][110111][101111][1111111]]
The representation of the complementary Fibonacci codes starts:
[0] [1]
[1] [2]
[2] [2, 6]
[3] [2, 6, 30]
[4] [2, 6, 30, 10, 210]
[5] [2, 6, 30, 10, 210, 42, 70, 2310]
[6] [2, 6, 30, 10, 210, 42, 70, 2310, 14, 330, 462, 770, 30030]
[7] [2, 6, 30, 10, 210, 42, 70, 2310, 14, 330, 462, 770, 30030, 66, 110, 2730, 154, 4290, 6006, 10010, 510510]
		

Crossrefs

The diagonal is A002110 (primorial numbers).

Cf. A309896, A327990, A000045, A001924.

Programs

SageMath

@cached_function
def FibonacciCodes(n):
    if n == 0 : return [[]]
    if n == 1 : return [[1]]
    A = [c.conjugate() for c in Compositions(n) if not(1 in c)]
    return FibonacciCodes(n-1) + [[2-i for i in a] for a in A]
def A327991row(n):
    P = Primes()
    M = lambda C: mul(P[i]^c for (i, c) in enumerate(C))
    return [M(c) for c in FibonacciCodes(n)]
for n in (0..7): print(A327991row(n))