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 A188866 #54 Sep 14 2021 02:49:46
%S A188866 2,4,3,8,7,4,16,17,10,5,32,41,26,13,6,64,99,68,35,16,7,128,239,178,95,
%T A188866 44,19,8,256,577,466,259,122,53,22,9,512,1393,1220,707,340,149,62,25,
%U A188866 10,1024,3363,3194,1931,950,421,176,71,28,11,2048,8119,8362,5275,2658,1193,502,203,80,31,12
%N A188866 T(n,k) is the number of n X k binary arrays without the pattern 0 1 diagonally, vertically or antidiagonally.
%C A188866 Number of 0..n strings of length k and adjacent elements differing by one or less. (See link for bijection.) Equivalently, number of base (n+1) k digit numbers with adjacent digits differing by one or less. - _Andrew Howroyd_, Mar 30 2017
%C A188866 All rows are linear recurrences with constant coefficients. See PARI script to obtain generating functions. - _Andrew Howroyd_, Apr 15 2017
%C A188866 Equivalently, the number of walks of length k-1 on the path graph P_{n+1} with a loop added at each vertex. - _Pontus von Brömssen_, Sep 08 2021
%H A188866 R. H. Hardin, <a href="/A188866/b188866.txt">Table of n, a(n) for n = 1..1741</a>
%H A188866 Andrew Howroyd, <a href="/A188866/a188866.txt">Bijection with 0..n strings of length k</a>.
%H A188866 Arnold Knopfmacher, Toufik Mansour, Augustine Munagi and Helmut Prodinger, <a href="http://arxiv.org/abs/0809.0551">Smooth words and Chebyshev polynomials</a>, arXiv:0809.0551v1 [math.CO], 2008.
%F A188866 Empirical: T(n,1) = n + 1.
%F A188866 Empirical: T(n,2) = 3*n + 1.
%F A188866 Empirical: T(n,3) = 9*n - 1.
%F A188866 Empirical: T(n,4) = 27*n - 13 for n > 1.
%F A188866 Empirical: T(n,5) = 81*n - 65 for n > 2.
%F A188866 Empirical: T(n,6) = 243*n - 265 for n > 3.
%F A188866 Empirical: T(n,7) = 729*n - 987 for n > 4.
%F A188866 Empirical: T(n,8) = 2187*n - 3495 for n > 5.
%F A188866 Empirical: T(1,k) = 2*T(1,k-1).
%F A188866 Empirical: T(2,k) = 2*T(2,k-1) + T(2,k-2).
%F A188866 Empirical: T(3,k) = 3*T(3,k-1) - T(3,k-2).
%F A188866 Empirical: T(4,k) = 3*T(4,k-1) - 2*T(4,k-3).
%F A188866 Empirical: T(5,k) = 4*T(5,k-1) - 3*T(5,k-2) - T(5,k-3).
%F A188866 Empirical: T(6,k) = 4*T(6,k-1) - 2*T(6,k-2) - 4*T(6,k-3) + T(6,k-4).
%F A188866 Empirical: T(7,k) = 5*T(7,k-1) - 6*T(7,k-2) - T(7,k-3) + 2*T(7,k-4).
%F A188866 Empirical: T(8,k) = 5*T(8,k-1) - 5*T(8,k-2) - 5*T(8,k-3) + 5*T(8,k-4) + T(8,k-5).
%e A188866 Table starts:
%e A188866 2 4 8 16 32 64 128 256 512 1024 2048 4096 8192 16384
%e A188866 3 7 17 41 99 239 577 1393 3363 8119 19601 47321 114243 275807
%e A188866 4 10 26 68 178 466 1220 3194 8362 21892 57314 150050 392836 1028458
%e A188866 5 13 35 95 259 707 1931 5275 14411 39371 107563 293867 802859 2193451
%e A188866 6 16 44 122 340 950 2658 7442 20844 58392 163594 458356 1284250 3598338
%e A188866 7 19 53 149 421 1193 3387 9627 27383 77923 221805 631469 1797957 5119593
%e A188866 8 22 62 176 502 1436 4116 11814 33942 97582 280676 807574 2324116 6689624
%e A188866 9 25 71 203 583 1679 4845 14001 40503 117263 339699 984515 2854281 8277153
%e A188866 10 28 80 230 664 1922 5574 16188 47064 136946 398746 1161634 3385486 9869934
%e A188866 11 31 89 257 745 2165 6303 18375 53625 156629 457795 1338779 3916897 11463989
%e A188866 Some solutions for 5 X 3:
%e A188866 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1
%e A188866 1 1 1 0 0 1 0 1 1 1 1 1 0 0 0 1 0 0 1 0 1
%e A188866 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0
%e A188866 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0
%e A188866 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
%t A188866 rows = 11; rowGf[n_, x_] = 1 + (x*(n - (3*n + 2)*x) + (2*x^2)*(1 + ChebyshevU[n-1, (1-x)/(2*x)])/ChebyshevU[n, (1-x)/(2*x)])/(1-3*x)^2;
%t A188866 row[n_] := rowGf[n+1, x] + O[x]^(rows+1) // CoefficientList[#, x]& // Rest; T = Array[row, rows]; Table[T[[n-k+1, k]], {n, 1, rows}, {k, n, 1, -1}] // Flatten (* _Jean-François Alcover_, Oct 07 2017, after _Andrew Howroyd_ *)
%o A188866 (PARI) \\ from Knopfmacher et al.
%o A188866 RowGf(k, x='x) = my(z=(1-x)/(2*x)); 1 + (x*(k-(3*k+2)*x) + (2*x^2)*(1+polchebyshev(k-1, 2, z))/polchebyshev(k, 2, z))/(1-3*x)^2;
%o A188866 T(n,k) = {polcoef(RowGf(n+1) + O(x*x^k),k)}
%o A188866 for(n=1, 10, print(Vec(RowGf(n+1) + O(x^11)))) \\ _Andrew Howroyd_, Apr 15 2017 [updated Mar 13 2021]
%Y A188866 Columns 2..8 are A016777, A017257(n-1), A188861-A188865.
%Y A188866 Rows 2..31 are A001333(n+1), A126358, A057960(n+1), A126360, A002714, A126362-A126386.
%Y A188866 Main diagonal is A188860.
%Y A188866 Cf. A276562, A220062.
%K A188866 nonn,tabl
%O A188866 1,1
%A A188866 _R. H. Hardin_, Apr 12 2011