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
KEYWORD
AUTHOR
Gary W. Adamson, Feb 22 2009
EXTENSIONS
Edited by G. C. Greubel, Jan 11 2022
STATUS
approved