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 A371692 #9 Apr 28 2024 11:03:34
%S A371692 1,1,2,1,2,2,1,2,4,3,1,2,4,6,6,1,2,4,8,11,9,1,2,4,8,14,19,15,1,2,4,8,
%T A371692 16,27,35,30,1,2,4,8,16,30,51,61,54,1,2,4,8,16,32,59,96,111,97,1,2,4,
%U A371692 8,16,32,62,115,183,200,189,1,2,4,8,16,32,64,123
%N A371692 Table(n,k) of binary strings of length n which have the same number of k long 0...00 and 0...01 substrings, where n>=0 and k>=2, read by downwards antidiagonals.
%C A371692 To clarify the substrings, k long '0...00' means k consecutive zeros, and k long '0...01' means k-1 consecutive zeros follow by a one.
%e A371692 Table begins:
%e A371692 n\k | 2 3 4 5 6 7 8 9 10
%e A371692 ----+----------------------------------------------------------------------
%e A371692 0 | 1, 1, 1, 1, 1, 1, 1, 1, 1
%e A371692 1 | 2, 2, 2, 2, 2, 2, 2, 2, 2
%e A371692 2 | 2, 4, 4, 4, 4, 4, 4, 4, 4
%e A371692 3 | 3, 6, 8, 8, 8, 8, 8, 8, 8
%e A371692 4 | 6, 11, 14, 16, 16, 16, 16, 16, 16
%e A371692 5 | 9, 19, 27, 30, 32, 32, 32, 32, 32
%e A371692 6 | 15, 35, 51, 59, 62, 64, 64, 64, 64
%e A371692 7 | 30, 61, 96, 115, 123, 126, 128, 128, 128
%e A371692 8 | 54, 111, 183, 224, 243, 251, 254, 256, 256
%e A371692 9 | 97, 200, 345, 436, 480, 499, 507, 510, 512
%e A371692 10 | 189, 369, 655, 851, 948, 992, 1011, 1019, 1022
%e A371692 11 | 360, 676, 1244, 1657, 1872, 1972, 2016, 2035, 2043
%e A371692 12 | 675, 1256, 2363, 3231, 3699, 3920, 4020, 4064, 4083
%e A371692 13 | 1304, 2337, 4500, 6300, 7305, 7792, 8016, 8116, 8160
%e A371692 14 | 2522, 4392, 8570, 12287, 14431, 15491, 15984, 16208, 16308
%e A371692 15 | 4835, 8273, 16347, 23966, 28508, 30793, 31872, 32368, 32592
%e A371692 16 | 9358, 15686, 31218, 46762, 56319, 61215, 63555, 64640, 65136
%e A371692 17 | 18193, 29837, 59678, 91250, 111266, 121692, 126729, 129088, 130176
%e A371692 18 | 35269, 57038, 114236, 178107, 219828, 241919, 252703, 257795, 260160
%e A371692 19 | 68568, 109362, 218905, 347709, 434338, 480930, 503900, 514825, 519936
%t A371692 l0[k_] := l0[k] = ConstantArray[0, k];
%t A371692 l1[k_] := l1[k] = ConstantArray[0, k - 1]~Join~{1};
%t A371692 tup[n_] := Tuples[{0, 1}, n];
%t A371692 cou[lst_List, k_] := Count[lst, l0[k]] == Count[lst, l1[k]];
%t A371692 par[lst_List, k_] := Partition[lst, k, 1];
%t A371692 a[n_, k_] := a[n, k] = Map[cou[#, k] &, Map[par[#, k] &, tup[n]]] // Boole // Total;
%t A371692 (* Data *)Table[a[n, k - n], {k, 2, 13}, {n, 0, k - 2}] // Flatten
%t A371692 (* Table *)Monitor[Table[a[n, k], {n, 0, 19}, {k, 2, 10}] // TableForm, {n, k}]
%Y A371692 Cf. A163493 (Column 1), A164137 (Column 2), A164147 (Column 3), A164178 (Column 4).
%K A371692 nonn,tabl
%O A371692 1,3
%A A371692 _Robert P. P. McKone_, Apr 03 2024