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

Author

Gary W. Adamson, Feb 22 2009

Keywords

• nonn
• easy
• tabl

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)

Examples

			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.
		

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.

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.

Programs

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

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 *)

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

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)

Extensions

Edited by G. C. Greubel, Jan 11 2022