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

1,3

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.

Links

Table of n, a(n) for n=1..74.

Example

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

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}]

Crossrefs

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

Sequence in context: A107027 A355395 A107030 * A271362 A354555 A263643

Adjacent sequences: A371689 A371690 A371691 * A371693 A371694 A371695

Keyword

nonn,tabl

Author

Robert P. P. McKone, Apr 03 2024

Status

approved