A157103 Array A(n, k) = Fibonacci(n+1, k), with A(n, 0) = A(n, n) = 1, read by antidiagonals.

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 5, 3, 1, 1, 5, 12, 10, 4, 1, 1, 8, 29, 33, 17, 5, 1, 1, 13, 70, 109, 72, 26, 6, 1, 1, 21, 169, 360, 305, 135, 37, 7, 1, 1, 34, 408, 1189, 1292, 701, 228, 50, 8, 1, 1, 55, 985, 3927, 5473, 3640, 1405, 357, 65, 9, 1

Offset

0,8

Comments

From Michael A. Allen, Mar 30 2023: (Start)

Column k is the k-metallonacci sequence for k > 0.

T(n,k) is, for n > 0 and k > 0, the number of tilings of an n-board (a board with dimensions n X 1) using unit squares and dominoes (with dimensions 2 X 1) if there are k kinds of squares available. (End)

Links

G. C. Greubel, Antidiagonals n = 0..50, flattened

Michael A. Allen and Kenneth Edwards, Fence tiling derived identities involving the metallonacci numbers squared or cubed, Fib. Q. 60:5 (2022) 5-17.

Michelle Rudolph-Lilith, On the Product Representation of Number Sequences, with Application to the Fibonacci Family, arXiv preprint arXiv:1508.07894 [math.NT], 2015. See Table 3.

Formula

A(n, k) = Fibonacci(n+1, k), with A(n, 0) = A(n, n) = 1 (array).

A(n, 1) = A000045(n+1).

T(n, k) = k*T(n-1, k) + T(n-2, k) with T(n, 0) = T(n, n) = 1 (triangle).

From G. C. Greubel, Jan 11 2022: (Start)

T(n, k) = Fibonacci(n-k+1, k), with T(n, 0) = T(n, n) = 1.

T(2*n, n) = A084845(n) for n >= 1, with T(0, 0) = 1.

T(2*n+1, n+1) = A084844(n). (End)

Example

Array begins:
  1,  1,   1,    1,     1,     1,      1,      1, ... (A000012);
  1,  1,   2,    3,     4,     5,      6,      7, ... (A000027);
  1,  2,   5,   10,    17,    26,     37,     50, ... (A002522);
  1,  3,  12,   33,    72,   135,    228,    357, ...;
  1,  5,  29,  109,   305,   701,   1405,   2549, ...;
  1,  8,  70,  360,  1292,  3640,   8658,  18200, ...;
  1, 13, 169, 1189,  5473, 18901,  53353, 129949, ...;
  1, 21, 408, 3927, 23184, 98145, 328776, 927843, ...;
  ...
First few rows of the triangle:
  1;
  1,   1;
  1,   1,    1;
  1,   2,    2,     1;
  1,   3,    5,     3,     1;
  1,   5,   12,    10,     4,     1;
  1,   8,   29,    33,    17,     5,     1;
  1,  13,   70,   109,    72,    26,     6,     1;
  1,  21,  169,   360,   305,   135,    37,     7,    1;
  1,  34,  408,  1189,  1292,   701,   228,    50,    8,   1;
  1,  55,  985,  3927,  5473,  3640,  1405,   357,   65,   9,   1;
  1,  89, 2378, 12970, 23184, 18901,  8658,  2549,  528,  82,  10,  1;
  1, 144, 5741, 42837, 98209, 98145, 53353, 18200, 4289, 747, 101, 11, 1;
  ...
Example: Column 3 = (1, 3, 10, 33, 109, 360, ...) = A006190.

Maple

A157103 := proc(n, k)
    if k = 0 then
        1;
    else
        mul(k-2*I*cos(l*Pi/(n+1)), l=1..n) ;
        combine(%, trig) ;
        round(%) ;
    end if;
end proc:
seq( seq(A157103(d-k, k), k=0..d), d=0..12) ; # R. J. Mathar, Feb 27 2023

Mathematica

(* First program *)
T[_, 0]=1; T[n_, n_]=1; T[_, _]=0;
T[n_, k_] /; 0 <= k <= n := k T[n-1, k] + T[n-2, k];
Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* Jean-François Alcover, Aug 07 2018 *)
(* Second program *)
T[n_, k_]:= If[k==0 || k==n, 1, Fibonacci[n-k+1, k]];
Table[T[n, k], {n, 0, 15}, {k, 0, n}]//Flatten (* G. C. Greubel, Jan 11 2022 *)

Prog

(Magma)
A157103:= func< n, k | k eq 0 or k eq n select 1 else Evaluate(DicksonSecond(n, -1), k) >;
[A157103(n-k, k): k in [0..n], n in [0..15]]; // G. C. Greubel, Jan 11 2022
(Sage)
def A157103(n, k): return 1 if (k==0 or k==n) else lucas_number1(n+1, k, -1)
flatten([[A157103(n-k, k) for k in (0..n)] for n in (0..10)]) # G. C. Greubel, Jan 11 2022

Crossrefs

Columns k=1 to 9: A000045, A000129, A006190, A001076, A052918, A005668, A054413, A041025, A099371.

Cf. A084844, A084845.

Essentially the transpose of A073133, A172236, A352361.

Sequence in context: A132044 A034327 A034254 * A135966 A292741 A356802

Adjacent sequences: A157100 A157101 A157102 * A157104 A157105 A157106

Keyword

nonn,easy,tabl

Author

Gary W. Adamson, Feb 22 2009

Extensions

Edited by G. C. Greubel, Jan 11 2022

Status

approved