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

%I A327991 #10 Oct 09 2019 02:31:35
%S A327991 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,
%T A327991 42,70,2310,14,330,462,770,30030,2,6,30,10,210,42,70,2310,14,330,462,
%U A327991 770,30030,66,110,2730,154,4290,6006,10010,510510
%N A327991 The complementary Fibonacci codes. Irregular triangle T(n, k) with n >= 0 and 0 <= k < A000045(n+1).
%C A327991 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.
%C A327991 The complementary Fibonacci codes are represented here through
%C A327991     T(n, k) = Product_{j=0..m} p(j)^c(j),
%C A327991 where p(j) is the j-th prime number, c = CF(n, k) and m = length(CF(n, k)).
%e A327991 The complementary Fibonacci codes start:
%e A327991 [0] [[]]
%e A327991 [1] [[1]]
%e A327991 [2] [[1][11]]
%e A327991 [3] [[1][11][111]]
%e A327991 [4] [[1][11][111][101][1111]]
%e A327991 [5] [[1][11][111][101][1111][1101][1011][11111]]
%e A327991 [6] [[1][11][111][101][1111][1101][1011][11111][1001][11101][11011][10111][111111]]
%e A327991 [7] [[1][11][111][101][1111][1101][1011][11111][1001][11101][11011][10111][111111] [11001][10101][111101][10011][111011][110111][101111][1111111]]
%e A327991 The representation of the complementary Fibonacci codes starts:
%e A327991 [0] [1]
%e A327991 [1] [2]
%e A327991 [2] [2, 6]
%e A327991 [3] [2, 6, 30]
%e A327991 [4] [2, 6, 30, 10, 210]
%e A327991 [5] [2, 6, 30, 10, 210, 42, 70, 2310]
%e A327991 [6] [2, 6, 30, 10, 210, 42, 70, 2310, 14, 330, 462, 770, 30030]
%e A327991 [7] [2, 6, 30, 10, 210, 42, 70, 2310, 14, 330, 462, 770, 30030, 66, 110, 2730, 154, 4290, 6006, 10010, 510510]
%o A327991 (SageMath)
%o A327991 @cached_function
%o A327991 def FibonacciCodes(n):
%o A327991     if n == 0 : return [[]]
%o A327991     if n == 1 : return [[1]]
%o A327991     A = [c.conjugate() for c in Compositions(n) if not(1 in c)]
%o A327991     return FibonacciCodes(n-1) + [[2-i for i in a] for a in A]
%o A327991 def A327991row(n):
%o A327991     P = Primes()
%o A327991     M = lambda C: mul(P[i]^c for (i, c) in enumerate(C))
%o A327991     return [M(c) for c in FibonacciCodes(n)]
%o A327991 for n in (0..7): print(A327991row(n))
%Y A327991 The diagonal is A002110 (primorial numbers).
%Y A327991 Cf. A309896, A327990, A000045, A001924.
%K A327991 nonn,tabf
%O A327991 0,2
%A A327991 _Peter Luschny_, Oct 09 2019