The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A327992 The binary Fibonacci compositions. Irregular triangle with n >= 0 where the length of row n is Fibonacci(n) for n > 0. 3
 1, 11, 111, 101, 1111, 1101, 1011, 11111, 1001, 11101, 11011, 10111, 111111, 11001, 10101, 10011, 111101, 111011, 110111, 101111, 1111111, 10001, 111001, 110101, 101101, 110011, 101011, 100111, 1111101, 1111011, 1110111, 1101111, 1011111, 11111111 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Taking up an idea of Cayley the binary Fibonacci compositions are defined as the conjugates of the compositions of n + 1 which do not have a part '1'. a(0) = 1 by convention and for n > 0 the representation of the composition c is given by Sum_{c} (2 - c[j])*2^j, where the c[j] are the parts of the composition c. With this interpretation the sequence is a permutation of the positive odd numbers (A005408). REFERENCES A. Cayley, Theorems in Trigonometry and on Partitions, Messenger of Mathematics, 5 (1876), pp. 164, 188. Also in Mathematical Papers Vol. 10, n. 634, p. 16. LINKS Peter Luschny, Table of n, a(n) for row 0..19 FORMULA The number of zeros in all binary Fibonacci compositions of n equal the number of elements in all subsets of {1, 2, ..., n} with no consecutive integers. (For example, the number of zeros in row 7 (see the triangle below) is 20 = A001629(6).) EXAMPLE The triangle starts:  [    1]  [   11]  [  111]  [  101,   1111]  [ 1101,   1011,  11111]  [ 1001,  11101,  11011,  10111, 111111]  [11001,  10101,  10011, 111101, 111011, 110111, 101111, 1111111]  [10001, 111001, 110101, 101101, 110011, 101011, 100111, 1111101, 1111011, 1110111, 1101111, 1011111, 11111111] . For instance, to compute T(7, 2) start with the composition [2, 3, 3]. Then take the conjugate, normalize the parts with 2 - c[j] and then represent the digits as an integer. The steps are: [2, 3, 3] -> [1, 1, 2, 1, 2, 1] -> [1, 1, 0, 1, 0, 1] -> 110101 = T(7, 2). PROG (SageMath) import functools def alpha(P, Q): # order of compositions     if len(P) < len(Q): return int(-1)     if len(P) == len(Q):         for i in range(len(P)):             if P[i] < Q[i]: return int(-1)             if P[i] > Q[i]: return int(1)             return int(0)     return int(0) def compositions(n):     A = [c.conjugate() for c in Compositions(n+1) if not(1 in c)]     B = [[2-i for i in a] for a in A]     sorted(B, key = functools.cmp_to_key(alpha))     return B def Int(c): # convert to decimal integer representation     s = ""     for t in c: s += str(t)     return Integer(s) if s else 1 def A327992row(n):     if n == 0: return      return [Int(c) for c in compositions(n)] for n in (0..8): print(A327992row(n)) CROSSREFS Cf. A000045, A001629, A327993 (row sums). Sequence in context: A004287 A061493 A093788 * A204847 A098759 A273977 Adjacent sequences:  A327989 A327990 A327991 * A327993 A327994 A327995 KEYWORD nonn,tabf AUTHOR Peter Luschny, Oct 12 2019 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified September 24 20:13 EDT 2020. Contains 337321 sequences. (Running on oeis4.)