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 A073133 #59 Jan 10 2025 01:16:32
%S A073133 1,1,1,1,2,2,1,3,5,3,1,4,10,12,5,1,5,17,33,29,8,1,6,26,72,109,70,13,1,
%T A073133 7,37,135,305,360,169,21,1,8,50,228,701,1292,1189,408,34,1,9,65,357,
%U A073133 1405,3640,5473,3927,985,55,1,10,82,528,2549,8658,18901,23184,12970,2378,89
%N A073133 Table by antidiagonals of T(n,k) = n*T(n,k-1) + T(n,k-2) starting with T(n,1) = 1.
%C A073133 Columns of the array are generated from Fibonacci polynomials f(x). They are: (1), (x), (x^2 + 1), (x^3 + 2x), (x^4 + 3x^2 + 1), (x^5 + 4x^3 + 3x), (x^6 + 5x^4 + 6x^2 +1), ... If column headings start 0, 1, 2, ... then the terms in the n-th column are generated from the n-th degree Fibonacci polynomial. For example, column 5 (8, 70, 360, ...) is generated from f(x), x = 1,2,3,...; fifth-degree polynomial x^5 + 4x^3 + 3x; e.g., f(2) = 70 = 2^5 + 4*8 + 3*2. - _Gary W. Adamson_, Apr 02 2006
%C A073133 The ratio of two consecutive entries of the sequence in the n-th row approaches (n + sqrt(n^2 + 4))/2. Example: The sequence beginning (1, 3, 10, 33, ...) tends to 3.302775... = (3 + sqrt(13))/2. - _Gary W. Adamson_, Aug 12 2013
%C A073133 As to the array sequences, (n+1)-th sequence is the INVERT transform of the n-th sequence. - _Gary W. Adamson_, Aug 20 2013
%C A073133 The array can be extended infinitely above the Fibonacci row by taking successive INVERTi transforms, resulting in:
%C A073133 ...
%C A073133 1, -2, 5, -12, 29, -70, ...
%C A073133 1, -1, 2, -3, 5, -8, ...
%C A073133 l, 0, 1, 0, 1, 0, ...
%C A073133 1, 1, 2, 3, 5, 8, ...
%C A073133 1, 2, 5, 12, 29, 70, ...
%C A073133 ...
%C A073133 This results in an infinite array in which sequences above the (1, 0, 1, 0, ...) are reflections of the sequences below, except for the alternate signs. Any sequence in the (+ sign) row starting (1, n, ...) is the (2*n-th) INVERT transform of the same sequence but with alternate signs. Example: (1, 2, 5, 12, ...) is the (2*2) = fourth INVERT transform of (1, -2, 5, -12, ...) by inspection. Conjecture: This "reflection" principle will result from taking successive INVERT transforms of any aerated sequence starting 1, ... and with positive signs. Likewise, the rows above the aerated sequence are successive INVERTi transforms of the aerated sequence. - _Gary W. Adamson_, Jul 14 2019
%C A073133 From _Michael A. Allen_, Feb 21 2023: (Start)
%C A073133 Row n is the n-metallonacci sequence.
%C A073133 T(n,k) 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 A073133 G. C. Greubel, <a href="/A073133/b073133.txt">Antidiagonals n = 1..100, flattened</a>
%H A073133 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 A073133 T(n, k) = A073134(n, k) + 2*A073135(n, k-2) = Sum_{j=0..k-1} abs(A049310(k-1, j)*n^j).
%F A073133 T(n,k) = [[0,1; 1,n]^{k+1}]_{1,1}, n,k in {1,2,...}. - _L. Edson Jeffery_, Sep 23 2012
%F A073133 G.f. for row n: x/(1-n*x-x^2). - _L. Edson Jeffery_, Aug 28 2013
%e A073133 Table begins:
%e A073133 1, 1, 2, 3, 5, 8, 13, ...
%e A073133 1, 2, 5, 12, 29, 70, 169, ...
%e A073133 1, 3, 10, 33, 109, 360, 1189, ...
%e A073133 1, 4, 17, 72, 305, 1292, 5473, ... etc.
%p A073133 A073133 := proc(n,k)
%p A073133 option remember;
%p A073133 if k <= 1 then
%p A073133 k;
%p A073133 else
%p A073133 n*procname(n,k-1)+procname(n,k-2) ;
%p A073133 end if;
%p A073133 end proc:
%p A073133 seq(seq( A073133(d-k,k),k=1..d-1),d=2..13) ; # _R. J. Mathar_, Aug 16 2019
%t A073133 T[n_, 1]:= 1; T[n_, k_]:= T[n, k] = If[k<0, 0, n*T[n, k-1] + T[n, k-2]]; Table[T[n-k+1, k], {n, 15}, {k, n}]//Flatten (* _G. C. Greubel_, Aug 12 2019 *)
%o A073133 (PARI) T(n,k) = if(k==1, 1, k<0, 0, n*T(n,k-1)+T(n,k-2));
%o A073133 for(n=1,15, for(k=1,n, print1(T(n-k+1,k), ", "))) \\ _G. C. Greubel_, Aug 12 2019
%o A073133 (Sage)
%o A073133 def T(n, k):
%o A073133 if (k<0): return 0
%o A073133 elif (k==1): return 1
%o A073133 else: return n*T(n, k-1) + T(n, k-2)
%o A073133 [[T(n-k+1, k) for k in (1..n)] for n in (1..15)] # _G. C. Greubel_, Aug 12 2019
%o A073133 (GAP)
%o A073133 T:= function(n,k)
%o A073133 if k<0 then return 0;
%o A073133 elif k=1 then return 1;
%o A073133 else return n*T(n,k-1) + T(n,k-2);
%o A073133 fi;
%o A073133 end;
%o A073133 Flat(List([1..15], n-> List([1..n], k-> T(n-k+1,k) ))); # _G. C. Greubel_, Aug 12 2019
%Y A073133 Rows are A000045, A000129, A006190, A001076, A052918, A005668, A054413, A041025, A099371, A041041, A049666, A041061, A140455, A041085, etc.
%Y A073133 Columns are A000012, A000027, A002522, A054602, A057721, A124152, etc.
%Y A073133 Different from A081572.
%K A073133 nonn,tabl,easy
%O A073133 1,5
%A A073133 _Henry Bottomley_, Jul 16 2002