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.

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, 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, 8, 16, 32, 62, 115, 183, 200, 189, 1, 2, 4, 8, 16, 32, 64, 123

Offset: 1

Author

Robert P. P. McKone, Apr 03 2024

Keywords

• nonn
• tabl

Comments

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.

Examples

			Table begins:
n\k |     2       3       4       5       6       7       8       9      10
----+----------------------------------------------------------------------
 0  |     1,      1,      1,      1,      1,      1,      1,      1,      1
 1  |     2,      2,      2,      2,      2,      2,      2,      2,      2
 2  |     2,      4,      4,      4,      4,      4,      4,      4,      4
 3  |     3,      6,      8,      8,      8,      8,      8,      8,      8
 4  |     6,     11,     14,     16,     16,     16,     16,     16,     16
 5  |     9,     19,     27,     30,     32,     32,     32,     32,     32
 6  |    15,     35,     51,     59,     62,     64,     64,     64,     64
 7  |    30,     61,     96,    115,    123,    126,    128,    128,    128
 8  |    54,    111,    183,    224,    243,    251,    254,    256,    256
 9  |    97,    200,    345,    436,    480,    499,    507,    510,    512
10  |   189,    369,    655,    851,    948,    992,   1011,   1019,   1022
11  |   360,    676,   1244,   1657,   1872,   1972,   2016,   2035,   2043
12  |   675,   1256,   2363,   3231,   3699,   3920,   4020,   4064,   4083
13  |  1304,   2337,   4500,   6300,   7305,   7792,   8016,   8116,   8160
14  |  2522,   4392,   8570,  12287,  14431,  15491,  15984,  16208,  16308
15  |  4835,   8273,  16347,  23966,  28508,  30793,  31872,  32368,  32592
16  |  9358,  15686,  31218,  46762,  56319,  61215,  63555,  64640,  65136
17  | 18193,  29837,  59678,  91250, 111266, 121692, 126729, 129088, 130176
18  | 35269,  57038, 114236, 178107, 219828, 241919, 252703, 257795, 260160
19  | 68568, 109362, 218905, 347709, 434338, 480930, 503900, 514825, 519936
		

Crossrefs

Cf. A163493 (Column 1), A164137 (Column 2), A164147 (Column 3), A164178 (Column 4).

Programs

Mathematica

l0[k_] := l0[k] = ConstantArray[0, k];
l1[k_] := l1[k] = ConstantArray[0, k - 1]~Join~{1};
tup[n_] := Tuples[{0, 1}, n];
cou[lst_List, k_] := Count[lst, l0[k]] == Count[lst, l1[k]];
par[lst_List, k_] := Partition[lst, k, 1];
a[n_, k_] := a[n, k] = Map[cou[#, k] &, Map[par[#, k] &, tup[n]]] // Boole // Total;
(* Data *)Table[a[n, k - n], {k, 2, 13}, {n, 0, k - 2}] // Flatten
(* Table *)Monitor[Table[a[n, k], {n, 0, 19}, {k, 2, 10}] // TableForm, {n, k}]