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

0,2

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)).

Links

Table of n, a(n) for n=0..53.

Example

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]

Prog

(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))

Crossrefs

The diagonal is A002110 (primorial numbers).

Cf. A309896, A327990, A000045, A001924.

Sequence in context: A068555 A167556 A221438 * A193322 A165460 A242649

Adjacent sequences: A327988 A327989 A327990 * A327992 A327993 A327994

Keyword

nonn,tabf

Author

Peter Luschny, Oct 09 2019

Status

approved