This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A352361 #53 Jan 05 2025 19:51:42
%S A352361 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,
%T A352361 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,
%U A352361 8,50,228,701,1292,1189,408,34,0,0,1,9,65,357,1405,3640,5473,3927,985,55,1
%N A352361 Array read by ascending antidiagonals. A(n, k) = Fibonacci(k, n), where Fibonacci(n, x) are the Fibonacci polynomials.
%C A352361 From _Michael A. Allen_, Mar 26 2023: (Start)
%C A352361 Row n is the n-metallonacci sequence for n>0.
%C A352361 A(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)
%H A352361 G. C. Greubel, <a href="/A352361/b352361.txt">Antidiagonals n = 0..50, flattened</a>
%H A352361 Michael A. Allen and Kenneth Edwards, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/60-5/allen.pdf">Fence tiling derived identities involving the metallonacci numbers squared or cubed</a>, Fib. Q. 60:5 (2022) 5-17.
%F A352361 A(n, k) = Sum_{j=0..floor((k-1)/2)} binomial(k-j-1, j)*n^(k-2*j-1).
%F A352361 A(n, k) = ((n + s)^k - (n - s)^k) / (2^k*s) where s = sqrt(n^2 + 4).
%F A352361 A(n, k) = [x^k] (x / (1 - n*x - x^2)).
%F A352361 A(n, k) = n^(k-1)*hypergeom([1 - k/2, 1/2 - k/2], [1 - k], -4/n^2) for n,k >= 1.
%F A352361 A(n, n) = T(2*n, n) = A084844(n).
%F A352361 From _G. C. Greubel_, Sep 29 2024: (Start)
%F A352361 T(n, k) = A(n-k, k) (antidiagonal triangle).
%F A352361 T(2*n+1, n+1) = A084845(n).
%F A352361 Sum_{k=0..n} T(n, k) = A304357(n) (row sums).
%F A352361 Sum_{k=0..n} (-1)^k*T(n, k) = (-1)*A304359(n). (End)
%e A352361 Array, A(n,k), starts:
%e A352361 n\k 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ...
%e A352361 -------------------------------------------------------------------------
%e A352361 [0] 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ... A000035;
%e A352361 [1] 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ... A000045;
%e A352361 [2] 0, 1, 2, 5, 12, 29, 70, 169, 408, 985, ... A000129;
%e A352361 [3] 0, 1, 3, 10, 33, 109, 360, 1189, 3927, 12970, ... A006190;
%e A352361 [4] 0, 1, 4, 17, 72, 305, 1292, 5473, 23184, 98209, ... A001076;
%e A352361 [5] 0, 1, 5, 26, 135, 701, 3640, 18901, 98145, 509626, ... A052918;
%e A352361 [6] 0, 1, 6, 37, 228, 1405, 8658, 53353, 328776, 2026009, ... A005668;
%e A352361 [7] 0, 1, 7, 50, 357, 2549, 18200, 129949, 927843, 6624850, ... A054413;
%e A352361 [8] 0, 1, 8, 65, 528, 4289, 34840, 283009, 2298912, 18674305, ... A041025;
%e A352361 [9] 0, 1, 9, 82, 747, 6805, 61992, 564733, 5144589, 46866034, ... A099371;
%e A352361 | | | | A054602 | A124152;
%e A352361 | | | A002522 A057721;
%e A352361 | | A001477;
%e A352361 | A000012;
%e A352361 A000004;
%e A352361 Antidiagonals, T(n, k), begin as:
%e A352361 0;
%e A352361 0, 1;
%e A352361 0, 1, 0;
%e A352361 0, 1, 1, 1;
%e A352361 0, 1, 2, 2, 0;
%e A352361 0, 1, 3, 5, 3, 1;
%e A352361 0, 1, 4, 10, 12, 5, 0;
%e A352361 0, 1, 5, 17, 33, 29, 8, 1;
%e A352361 0, 1, 6, 26, 72, 109, 70, 13, 0;
%e A352361 0, 1, 7, 37, 135, 305, 360, 169, 21, 1;
%p A352361 seq(seq(combinat:-fibonacci(k, n - k), k = 0..n), n = 0..11);
%t A352361 Table[Fibonacci[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten
%t A352361 (* or *)
%t A352361 A[n_, k_] := With[{s = Sqrt[n^2 + 4]}, ((n + s)^k - (n - s)^k) / (2^k*s)];
%t A352361 Table[Simplify[A[n, k]], {n, 0, 9}, {k, 0, 9}] // TableForm
%o A352361 (PARI)
%o A352361 A(n, k) = ([1, k; 1, k-1]^n)[2, 1] ;
%o A352361 export(A)
%o A352361 for(k = 0, 9, print(parvector(10, n, A(n - 1, k))))
%o A352361 (Magma)
%o A352361 A352361:= func< n, k | k le 1 select k else Evaluate(DicksonSecond(k-1, -1), n-k) >;
%o A352361 [A352361(n, k): k in [0..n], n in [0..13]]; // _G. C. Greubel_, Sep 29 2024
%o A352361 (SageMath)
%o A352361 def A352361(n, k): return lucas_number1(k,n-k,-1)
%o A352361 flatten([[A352361(n, k) for k in range(n+1)] for n in range(14)]) # _G. C. Greubel_, Sep 29 2024
%Y A352361 Other versions of this array are A073133, A157103, A172236.
%Y A352361 Rows n: A000035 (n=0), A000045 (n=1), A000129 (n=2), A006190 (n=3), A001076 (n=4), A052918 (n=5), A005668 (n=6), A054413 (n=7), A041025 (n=8), A099371 (n=9).
%Y A352361 Columns k: A000004 (k=0), A000012 (k=1), A001477 (k=2), A002522 (k=3), A054602 (k=4), A057721 (k=5), A124152 (k=6).
%Y A352361 Cf. A084844 (main diagonal), A352362 (Lucas polynomials), A350470 (Jacobsthal polynomials).
%Y A352361 Sums include: A304357 (row sums), A304359.
%Y A352361 Cf. A084845.
%K A352361 nonn,easy,tabl
%O A352361 0,13
%A A352361 _Peter Luschny_, Mar 18 2022