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 A092921 #108 Aug 04 2025 09:45:51
%S A092921 0,1,0,1,1,0,1,1,1,0,1,2,1,1,0,1,3,2,1,1,0,1,5,4,2,1,1,0,1,8,7,4,2,1,
%T A092921 1,0,1,13,13,8,4,2,1,1,0,1,21,24,15,8,4,2,1,1,0,1,34,44,29,16,8,4,2,1,
%U A092921 1,0,1,55,81,56,31,16,8,4,2,1,1,0,1,89,149,108,61,32,16,8,4,2,1,1,0
%N A092921 Array F(k, n) read by descending antidiagonals: k-generalized Fibonacci numbers in row k >= 1, starting (0, 1, 1, ...), for column n >= 0.
%C A092921 For all k >= 1, the k-generalized Fibonacci number F(k,n) satisfies the recurrence obtained by adding more terms to the recurrence of the Fibonacci numbers.
%C A092921 The number of tilings of an 1 X n rectangle with tiles of size 1 X 1, 1 X 2, ..., 1 X k is F(k,n).
%C A092921 T(k,n) is the number of 0-balanced ordered trees with n edges and height k (height is the number of edges from root to a leaf). - _Emeric Deutsch_, Jan 19 2007
%C A092921 Brlek et al. (2006) call this table "number of psp-polyominoes with flat bottom". - _N. J. A. Sloane_, Oct 30 2018
%H A092921 Alois P. Heinz, <a href="/A092921/b092921.txt">Antidiagonals n = 0..140, flattened</a>
%H A092921 Srecko Brlek, Andrea Frosini, Simone Rinaldi, and Laurent Vuillon, <a href="https://doi.org/10.37236/1041">Tilings by translation: enumeration by a rational language approach</a>, The Electronic Journal of Combinatorics, vol. 13, (2006). Table 1 is essentially this array. - _N. J. A. Sloane_, Jul 20 2014
%H A092921 Eric S. Egge, <a href="https://arxiv.org/abs/math/0109219">Restricted permutations related to Fibonacci numbers and k-generalized Fibonacci numbers</a>, arXiv:math/0109219 [math.CO], 2001.
%H A092921 Eric S. Egge, <a href="https://arxiv.org/abs/math/0307050">Restricted 3412-Avoiding Involutions</a>, arXiv:math/0307050 [math.CO], 2003.
%H A092921 Eric S. Egge and Toufik Mansour, <a href="https://arxiv.org/abs/math/0203226">Restricted permutations, Fibonacci numbers and k-generalized Fibonacci numbers</a>, arXiv:math/0203226 [math.CO], 2002.
%H A092921 Eric S. Egge and Toufik Mansour, <a href="https://arxiv.org/abs/math/0209255">231-avoiding involutions and Fibonacci numbers</a>, arXiv:math/0209255 [math.CO], 2002.
%H A092921 Nathaniel D. Emerson, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL9/Emerson/emerson6.html">A Family of Meta-Fibonacci Sequences Defined by Variable-Order Recursions</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.8.
%H A092921 Abraham Flaxman, Aram W. Harrow, and Gregory B. Sorkin, <a href="https://doi.org/10.37236/1761">Strings with Maximally Many Distinct Subsequences and Substrings</a>, Electronic J. Combinatorics 11 (1) (2004), Paper R8.
%H A092921 Ivan Flores, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/5-3/flores.pdf">k-Generalized Fibonacci numbers</a>, Fib. Quart., 5 (1967), 258-266.
%H A092921 H. Gabai, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/8-1/gabai.pdf">Generalized Fibonacci k-sequences</a>, Fib. Quart., 8 (1970), 31-38.
%H A092921 R. Kemp, <a href="http://dx.doi.org/10.1002/rsa.3240050111">Balanced ordered trees</a>, Random Structures and Alg., 5 (1994), pp. 99-121.
%H A092921 E. P. Miles jr., <a href="http://www.jstor.org/stable/2308649">Generalized Fibonacci numbers and associated matrices</a>, The Amer. Math. Monthly, 67 (1960) 745-752.
%H A092921 M. D. Miller, <a href="http://www.jstor.org/stable/2316316">On generalized Fibonacci numbers</a>, The Amer. Math. Monthly, 78 (1971) 1108-1109.
%H A092921 Michel Mollard, <a href="https://arxiv.org/abs/2507.16387">p-th order generalized Fibonacci cubes and maximal cubes in Fibonacci p-cubes</a>, arXiv:2507.16387 [math.CO], 2025. See p. 3.
%H A092921 Harold R. Parks and Dean C. Wills, <a href="https://arxiv.org/abs/2208.01224">Sum of k-bonacci Numbers</a>, arXiv:2208.01224 [math.CO], 2022. See p. 5.
%H A092921 Hsin-Po Wang and Chi-Wei Chin, <a href="https://arxiv.org/abs/2405.17499">On Counting Subsequences and Higher-Order Fibonacci Numbers</a>, arXiv:2405.17499 [cs.IT], 2024. See p. 2.
%F A092921 F(k,n) = F(k,n-1) + F(k,n-2) + ... + F(k,n-k); F(k,1) = 1 and F(k,n) = 0 for n <= 0.
%F A092921 G.f.: x/(1-Sum_{i=1..k} x^i).
%F A092921 F(k,n) = 2^(n-2) for 1 < n <= k+1. - _M. F. Hasler_, Apr 20 2018
%F A092921 F(k,n) = Sum_{j=0..floor(n/(k+1))} (-1)^j*((n - j*k) + j + delta(n,0))/(2*(n - j*k) + delta(n,0))*binomial(n - j*k, j)*2^(n-j*(k+1)), where delta denotes the Kronecker delta (see Corollary 3.2 in Parks and Wills). - _Stefano Spezia_, Aug 06 2022
%e A092921 From _Peter Luschny_, Apr 03 2021: (Start)
%e A092921 Array begins:
%e A092921 n = 0 1 2 3 4 5 6 7 8 9 10
%e A092921 -------------------------------------------------------------
%e A092921 [k=1, mononacci ] 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e A092921 [k=2, Fibonacci ] 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ...
%e A092921 [k=3, tribonacci] 0, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149, ...
%e A092921 [k=4, tetranacci] 0, 1, 1, 2, 4, 8, 15, 29, 56, 108, 208, ...
%e A092921 [k=5, pentanacci] 0, 1, 1, 2, 4, 8, 16, 31, 61, 120, 236, ...
%e A092921 [k=6] 0, 1, 1, 2, 4, 8, 16, 32, 63, 125, 248, ...
%e A092921 [k=7] 0, 1, 1, 2, 4, 8, 16, 32, 64, 127, 253, ...
%e A092921 [k=8] 0, 1, 1, 2, 4, 8, 16, 32, 64, 128, 255, ...
%e A092921 [k=9] 0, 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, ...
%e A092921 Note that the first parameter in F(k, n) refers to rows, and the second parameter refers to columns. This is always the case. Only the usual naming convention for the indices is not adhered to because it is common to call the row sequences k-bonacci numbers. (End)
%e A092921 .
%e A092921 From _Peter Luschny_, Aug 12 2015: (Start)
%e A092921 As a triangle counting compositions of n with largest part k:
%e A092921 [n\k]| [0][1] [2] [3] [4][5][6][7][8][9]
%e A092921 [0] | [0]
%e A092921 [1] | [0, 1]
%e A092921 [2] | [0, 1, 1]
%e A092921 [3] | [0, 1, 1, 1]
%e A092921 [4] | [0, 1, 2, 1, 1]
%e A092921 [5] | [0, 1, 3, 2, 1, 1]
%e A092921 [6] | [0, 1, 5, 4, 2, 1, 1]
%e A092921 [7] | [0, 1, 8, 7, 4, 2, 1, 1]
%e A092921 [8] | [0, 1, 13, 13, 8, 4, 2, 1, 1]
%e A092921 [9] | [0, 1, 21, 24, 15, 8, 4, 2, 1, 1]
%e A092921 For example for n=7 and k=3 we have the 7 compositions [3, 3, 1], [3, 2, 2], [3, 2, 1, 1], [3, 1, 3], [3, 1, 2, 1], [3, 1, 1, 2], [3, 1, 1, 1, 1]. (End)
%p A092921 F:= proc(k, n) option remember; `if`(n<2, n,
%p A092921 add(F(k, n-j), j=1..min(k,n)))
%p A092921 end:
%p A092921 seq(seq(F(k, d+1-k), k=1..d+1), d=0..12); # _Alois P. Heinz_, Nov 02 2016
%p A092921 # Based on the above function:
%p A092921 Arow := (k, len) -> seq(F(k, j), j = 0..len):
%p A092921 seq(lprint(Arow(k, 14)), k = 1..10); # _Peter Luschny_, Apr 03 2021
%t A092921 F[k_, n_] := F[k, n] = If[n<2, n, Sum[F[k, n-j], {j, 1, Min[k, n]}]];
%t A092921 Table[F[k, d+1-k], {d, 0, 12}, {k, 1, d+1}] // Flatten (* _Jean-François Alcover_, Jan 11 2017, translated from Maple *)
%o A092921 (PARI) F(k,n)=if(n<2,if(n<1,0,1),sum(i=1,k,F(k,n-i)))
%o A092921 (PARI) T(m,n)=!!n*(matrix(m,m,i,j,j==i+1||i==m)^(n+m-2))[1,m] \\ _M. F. Hasler_, Apr 20 2018
%o A092921 (PARI) F(k,n) = if(n==0,0, polcoeff(lift(Mod('x, Pol(vector(k+1,i, if(i==1,1,-1))))^(n+k-2)), k-1)); \\ _Kevin Ryde_, Jun 05 2020
%o A092921 (Sage)
%o A092921 # As a triangle of compositions of n with largest part k.
%o A092921 C = lambda n,k: Compositions(n, max_part=k, inner=[k]).cardinality()
%o A092921 for n in (0..9): [C(n,k) for k in (0..n)] # _Peter Luschny_, Aug 12 2015
%Y A092921 Columns converge to A166444: each column n converges to A166444(n) = 2^(n-2).
%Y A092921 Rows 1-8 are (shifted) A057427, A000045, A000073, A000078, A001591, A001592, A066178, A079262.
%Y A092921 Essentially a reflected version of A048887.
%Y A092921 See A048004 and A126198 for closely related arrays.
%Y A092921 Cf. A066099.
%K A092921 nonn,tabl
%O A092921 0,12
%A A092921 _Ralf Stephan_, Apr 17 2004