A352361 Array read by ascending antidiagonals. T(n, k) = F(k, n), where F are the Fibonacci polynomials.

0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 2, 2, 0, 0, 1, 3, 5, 3, 1, 0, 1, 4, 10, 12, 5, 0, 0, 1, 5, 17, 33, 29, 8, 1, 0, 1, 6, 26, 72, 109, 70, 13, 0, 0, 1, 7, 37, 135, 305, 360, 169, 21, 1, 0, 1, 8, 50, 228, 701, 1292, 1189, 408, 34, 0, 0, 1, 9, 65, 357, 1405, 3640, 5473, 3927, 985, 55, 1

Offset

0,13

Comments

From Michael A. Allen, Mar 26 2023: (Start)

Row n is the n-metallonacci sequence for n>0.

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

Links

Table of n, a(n) for n=0..77.

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

Formula

T(n, k) = Sum_{j=0..floor((k-1)/2)} binomial(k-j-1, j)*n^(k-2*j-1).

T(n, k) = ((n + s)^k - (n - s)^k) / (2^k*s) where s = sqrt(n^2 + 4).

T(n, k) = [x^k] (x / (1 - n*x - x^2)).

T(n, k) = n^(k-1)*hypergeom([1 - k/2, 1/2 - k/2], [1 - k], -4/n^2) for n,k >= 1.

Example

Array starts:
n\k 0, 1, 2,  3,   4,    5,     6,      7,       8,        9, ...
-------------------------------------------------------------------------
[0] 0, 1, 0,  1,   0,    1,     0,      1,       0,        1, ... A000035
[1] 0, 1, 1,  2,   3,    5,     8,     13,      21,       34, ... A000045
[2] 0, 1, 2,  5,  12,   29,    70,    169,     408,      985, ... A000129
[3] 0, 1, 3, 10,  33,  109,   360,   1189,    3927,    12970, ... A006190
[4] 0, 1, 4, 17,  72,  305,  1292,   5473,   23184,    98209, ... A001076
[5] 0, 1, 5, 26, 135,  701,  3640,  18901,   98145,   509626, ... A052918
[6] 0, 1, 6, 37, 228, 1405,  8658,  53353,  328776,  2026009, ... A005668
[7] 0, 1, 7, 50, 357, 2549, 18200, 129949,  927843,  6624850, ... A054413
[8] 0, 1, 8, 65, 528, 4289, 34840, 283009, 2298912, 18674305, ... A041025
[9] 0, 1, 9, 82, 747, 6805, 61992, 564733, 5144589, 46866034, ... A099371
    |  |  |  | A054602 |   A124152
    |  |  |  A002522   A057721
    |  |  A001477
    |  A000012
    A000004

Maple

seq(seq(combinat:-fibonacci(k, n - k), k = 0..n), n = 0..11);

Mathematica

Table[Fibonacci[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten
(* or *)
T[n_, k_] := With[{s = Sqrt[n^2 + 4]}, ((n + s)^k - (n - s)^k) / (2^k*s)];
Table[Simplify[T[n, k]], {n, 0, 9}, {k, 0, 9}] // TableForm

Prog

(PARI)
T(n, k) = ([1, k; 1, k-1]^n)[2, 1] ; export(T)
for(k = 0, 9, print(parvector(10, n, T(n - 1, k))))

Crossrefs

Other versions of this array are A073133, A157103, A172236.

Rows: A000035, A000045, A000129, A006190, A001076, A052918, A005668, A054413, A041025, A099371.

Columns: A000004, A000012, A001477, A002522, A054602, A057721, A124152.

Cf. A084844 (main diagonal), A352362 (Lucas polynomials), A350470 (Jacobsthal polynomials).

Sequence in context: A136438 A370063 A059848 * A036865 A242249 A125226

Adjacent sequences: A352358 A352359 A352360 * A352362 A352363 A352364

Keyword

nonn,easy,tabl

Author

Peter Luschny, Mar 18 2022

Status

approved