A309896 -id:A309896 - cp's OEIS frontend

A259475 Array read by antidiagonals: row n gives coefficients of Taylor series expansion of 1/F_{n+1}(t), where F_i(t) is a Fibonacci polynomial defined by F_0=1, F_1=1, F_{i+1} = F_i-t*F_{i-1}.

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 4, 1, 0, 1, 4, 8, 8, 1, 0, 1, 5, 13, 21, 16, 1, 0, 1, 6, 19, 40, 55, 32, 1, 0, 1, 7, 26, 66, 121, 144, 64, 1, 0, 1, 8, 34, 100, 221, 364, 377, 128, 1, 0, 1, 9, 43, 143, 364, 728, 1093, 987, 256, 1, 0, 1, 10, 53, 196, 560, 1288, 2380, 3280, 2584, 512, 1, 0

Offset: 0

N. J. A. Sloane, Jul 03 2015

			The first few antidiagonals are:
  1;
  1, 0;
  1, 1,  0;
  1, 2,  1,  0;
  1, 3,  4,  1,   0;
  1, 4,  8,  8,   1,   0;
  1, 5, 13, 21,  16,   1,  0;
  1, 6, 19, 40,  55,  32,  1, 0;
  1, 7, 26, 66, 121, 144, 64, 1, 0;
  ...
Square array starts:
  [0] 1, 0,  0,   0,    0,    0,     0,     0,      0,       0,       0, ...
  [1] 1, 1,  1,   1,    1,    1,     1,     1,      1,       1,       1, ...
  [2] 1, 2,  4,   8,   16,   32,    64,   128,    256,     512,    1024, ...
  [3] 1, 3,  8,  21,   55,  144,   377,   987,   2584,    6765,   17711, ...
  [4] 1, 4, 13,  40,  121,  364,  1093,  3280,   9841,   29524,   88573, ...
  [5] 1, 5, 19,  66,  221,  728,  2380,  7753,  25213,   81927,  266110, ...
  [6] 1, 6, 26, 100,  364, 1288,  4488, 15504,  53296,  182688,  625184, ...
  [7] 1, 7, 34, 143,  560, 2108,  7752, 28101, 100947,  360526, 1282735, ...
  [8] 1, 8, 43, 196,  820, 3264, 12597, 47652, 177859,  657800, 2417416, ...
  [9] 1, 9, 53, 260, 1156, 4845, 19551, 76912, 297275, 1134705, 4292145, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
Noureddine Chair, Explicit Computations for the Intersection Numbers on Grassmannians, and on the Space of Holomorphic Maps from CP¹ into G_r(Cⁿ), arXiv:hep-th/9808170, 1998. (Cf. Table 4).
G. Kreweras, Sur les éventails de segments, Cahiers du Bureau Universitaire de Recherche Opérationelle, Cahier no. 15, Paris, 1970, pp. 3-41.
G. Kreweras, Sur les éventails de segments, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #15 (1970), 3-41. [Annotated scanned copy]
Genki Shibukawa, New identities for some symmetric polynomials and their applications, arXiv:1907.00334 [math.CA], 2019.

The initial rows of the array are A000007, A000012, A000079, A001906, A003432, A005021, A094811, A094256.
A(n,n) gives A274969.
Cf. A309896.
A188843 is a variant without the first two rows and the first column, and the antidiagonals read in opposite direction.

F:= proc(n) option remember;
      `if`(n<2, 1, expand(F(n-1)-t*F(n-2)))
    end:
A:= (n, k)-> coeff(series(1/F(n+1), t, k+1), t, k):
seq(seq(A(d-k, k), k=0..d), d=0..12);  # Alois P. Heinz, Jul 04 2015

F[n_] := F[n] = If[n<2, 1, Expand[F[n-1] - t*F[n-2]]]; A[n_, k_] := SeriesCoefficient[1/F[n+1], { t, 0, k}]; Table[A[d-k, k], {d, 0, 12}, {k, 0, d}] // Flatten (* Jean-François Alcover, Feb 17 2016, after Alois P. Heinz *)

@cached_function
def F(n, k):
    if k <  0: return 0
    if k == 0: return 1
    return sum((-1)^j*binomial(n-1-j,j+1)*F(n,k-2-2*j) for j in (0..(n-2)/2))
def A(n, k): return F(n+1, 2*k)
print([A(n-k, k) for n in (0..11) for k in (0..n)]) # Peter Luschny, Aug 21 2019

Let F(n, k) = Sum_{j=0..(n-2)/2} (-1)^j*binomial(n-1-j, j+1)*F(n, k-2-2*j) for k > 0; F(n, 0) = 1 and F(n, k) = 0 if k < 0. Then A(n, k) = F(n+1, 2*k). See [Shibukawa] and A309896. - Peter Luschny, Aug 21 2019

More terms from Alois P. Heinz, Jul 04 2015

A327990 The Fibonacci Codes. Irregular triangle T(n, k) with n >= 0 and 0 <= k < A000045(n+1).

Original entry on oeis.org

0, 1, 3, 1, 7, 3, 1, 9, 15, 7, 3, 1, 19, 17, 31, 9, 15, 7, 3, 1, 21, 39, 35, 33, 63, 19, 17, 31, 9, 15, 7, 3, 1, 43, 41, 79, 37, 71, 67, 65, 127, 21, 39, 35, 33, 63, 19, 17, 31, 9, 15, 7, 3, 1

Offset: 0

Peter Luschny, Oct 08 2019

The Fibonacci codes are binary strings enumerated in an irregular triangle FC(n, k). The first few are shown below in the Example section.
The Fibonacci codes are for n > 1 defined recursively FC(n) = C(n) concatenated with FC(n-1), where C(n) are the conjugates of the compositions of n which do not have '1' as a part and the parts of which were reduced by 1. The recurrence is based in FC(0) = '' (empty string) and FC(1) = '0'.
The Fibonacci numbers are defined F(n) = A309896(2,n) = A000045(n+1) for n >= 0. Row FC(n) contains F(n) codes. A nonzero code is a code that does not consist entirely of zeros. The number of nonzero codes in row n is A001924(n-3) for n>=3.
Fibonacci codes are represented here through
    T(n, k) = Sum_{j=0..m} (c[j] + 1)*2^j,
where c = FC(n, k) and m = length(FC(n, k)).

			The Fibonacci codes start:
[0] [[]]
[1] [[0]]
[2] [[00][0]]
[3] [[000][00][0]]
[4] [[010][0000][000][00][0]]
[5] [[0010][0100][00000][010][0000][000][00][0]]
[6] [[0110][00010][00100][01000][000000][0010][0100][00000][010][0000][000][00][0]]
[7] [[00110][01010][000010][01100][000100][001000][010000][0000000][0110][00010][00100][01000][000000][0010][0100][00000][010][0000][000][00][0]]
The encoding of the Fibonacci codes start:
[0] [0]
[1] [1]
[2] [3, 1]
[3] [7, 3, 1]
[4] [9, 15, 7, 3, 1]
[5] [19, 17, 31, 9, 15, 7, 3, 1]
[6] [21, 39, 35, 33, 63, 19, 17, 31, 9, 15, 7, 3, 1]
[7] [43, 41, 79, 37, 71, 67, 65, 127, 21, 39, 35, 33, 63, 19, 17, 31, 9, 15, 7, 3, 1]

Cf. A309896, A000045, A001924.

@cached_function
def FibonacciCodes(n):
    if n == 0 : return [[]]
    if n == 1 : return [[0]]
    A = [c.conjugate() for c in Compositions(n) if not(1 in c)]
    B = [[i-1 for i in a] for a in A]
    return B + FibonacciCodes(n-1)
def A327990row(n):
    FC = FibonacciCodes(n)
    B = lambda C: sum((c+1)*2^i for (i, c) in enumerate(C))
    return [B(c) for c in FC]
for n in (0..6): print(A327990row(n))

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

Original entry on oeis.org

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

Peter Luschny, Oct 09 2019

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)).

			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]

The diagonal is A002110 (primorial numbers).

Cf. A309896, A327990, A000045, A001924.

@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))

cp's OEIS Frontend

A259475 Array read by antidiagonals: row n gives coefficients of Taylor series expansion of 1/F_{n+1}(t), where F_i(t) is a Fibonacci polynomial defined by F_0=1, F_1=1, F_{i+1} = F_i-t*F_{i-1}.

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A327990 The Fibonacci Codes. Irregular triangle T(n, k) with n >= 0 and 0 <= k < A000045(n+1).

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Author

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Comments

Examples

Crossrefs

Programs

SageMath

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

SageMath