A307796 -id:A307796 - cp's OEIS frontend

A303696 Number A(n,k) of binary words of length n with k times as many occurrences of subword 101 as occurrences of subword 010; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 2, 1, 2, 4, 1, 2, 4, 7, 1, 2, 4, 6, 12, 1, 2, 4, 6, 12, 21, 1, 2, 4, 6, 10, 20, 37, 1, 2, 4, 6, 10, 17, 38, 65, 1, 2, 4, 6, 10, 16, 28, 66, 114, 1, 2, 4, 6, 10, 16, 26, 49, 124, 200, 1, 2, 4, 6, 10, 16, 26, 42, 84, 224, 351, 1, 2, 4, 6, 10, 16, 26, 42, 70, 148, 424, 616

Offset: 0

Alois P. Heinz, Apr 28 2018

A(n,n) is the number of binary words of length n avoiding both subwords 101 and 010. A(4,4) = 10: 0000, 0001, 0011, 0110, 0111, 1000, 1001, 1100, 1110, 1111.

			Square array A(n,k) begins:
    1,   1,   1,   1,   1,   1,   1, ...
    2,   2,   2,   2,   2,   2,   2, ...
    4,   4,   4,   4,   4,   4,   4, ...
    7,   6,   6,   6,   6,   6,   6, ...
   12,  12,  10,  10,  10,  10,  10, ...
   21,  20,  17,  16,  16,  16,  16, ...
   37,  38,  28,  26,  26,  26,  26, ...
   65,  66,  49,  42,  42,  42,  42, ...
  114, 124,  84,  70,  68,  68,  68, ...
  200, 224, 148, 116, 110, 110, 110, ...
  351, 424, 263, 196, 178, 178, 178, ...

Alois P. Heinz, Antidiagonals for n = 0..200, flattened

Columns k=0-3 give: A005251(n+3), A164146, A303430, A307795.
Main diagonal gives A128588(n+1).
Cf. A000045, A307796.

b:= proc(n, t, h, c, k) option remember; `if`(abs(c)>k*n, 0,
     `if`(n=0, 1, b(n-1, [1, 3, 1][t], 2, c-`if`(h=3, k, 0), k)
                + b(n-1, 2, [1, 3, 1][h], c+`if`(t=3, 1, 0), k)))
    end:
A:= (n, k)-> b(n, 1$2, 0, min(k, n)):
seq(seq(A(n, d-n), n=0..d), d=0..14);

b[n_, t_, h_, c_, k_] := b[n, t, h, c, k] = If[Abs[c] > k n, 0, If[n == 0, 1, b[n - 1, {1, 3, 1}[[t]], 2, c - If[h == 3, k, 0], k] + b[n - 1, 2, {1, 3, 1}[[h]], c + If[t == 3, 1, 0], k]]];
A[n_, k_] := b[n, 1, 1, 0, Min[k, n]];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Mar 20 2020, from Maple *)

ceiling(A(n,n)/2) = A000045(n+1).

A164146 Number of binary strings of length n with equal numbers of 010 and 101 substrings.

Original entry on oeis.org

1, 2, 4, 6, 12, 20, 38, 66, 124, 224, 424, 788, 1502, 2838, 5438, 10386, 20004, 38508, 74516, 144264, 280216, 544736, 1061292, 2069596, 4042254, 7902294, 15466842, 30297422, 59404174, 116558270, 228876426, 449713994, 884199348, 1739434972, 3423770240, 6742430340

Offset: 0

R. H. Hardin, Aug 11 2009

			a(5) = 20: 00000, 00001, 00011, 00101, 00110, 00111, 01011, 01100, 01110, 01111, 10000, 10001, 10011, 10100, 11000, 11001, 11010, 11100, 11110, 11111. - _Alois P. Heinz_, Apr 16 2015

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 501 terms from R. H. Hardin)
Shalosh B. Ekhad and Doron Zeilberger, Automatic Solution of Richard Stanley's Amer. Math. Monthly Problem #11610 and ANY Problem of That Type, arXiv preprint arXiv:1112.6207, 2011. See subpages for rigorous derivations of g.f., recurrence, asymptotics for this sequence. [From _N. J. A. Sloane_, Apr 07 2012]

Cf. A118430, A255386, A260505, A260668, A260697, A303430.
Column k=1 of A303696.
Column k=0 of A307796.

CoefficientList[Series[-(4*x^4-2*x^3-2*x^2+x+Sqrt[(2*x-1)*(2*x^2-1)*(2*x^2-2*x+1)]) / ((x-1)*(2*x-1)*(2*x^2-1)),{x,0,33}],x] (* Stefano Spezia, Jul 31 2025 *)

G.f.: -(4*x^4-2*x^3-2*x^2+x+sqrt((2*x-1)*(2*x^2-1)*(2*x^2-2*x+1))) / ((x-1)*(2*x-1)*(2*x^2-1)). - Alois P. Heinz, Apr 16 2015

A284449 Number of n X 1 0..1 arrays with the number of 1's king-move adjacent to some 0 one less than the number of 0's adjacent to some 1.

Original entry on oeis.org

0, 0, 0, 1, 2, 6, 12, 28, 56, 119, 236, 481, 950, 1902, 3752, 7450, 14684, 29032, 57192, 112850, 222308, 438359, 863808, 1703239, 3357766, 6622471, 13061980, 25772503, 50859826, 100399602, 198235896, 391523612, 773453896, 1528361734, 3020781528, 5971996960

Offset: 0

R. H. Hardin, Mar 27 2017

nonn

Number of binary words of length n with exactly one occurrence of subword 101 more than occurrences of subword 010. a(5) = 6: 01101, 10101, 10110, 10111, 11011, 11101. - Alois P. Heinz, Apr 23 2018

			Both solutions for n=4
..0. .0
..1. .0
..0. .1
..0. .0

Alois P. Heinz, Table of n, a(n) for n = 0..3327 (first 210 terms from R. H. Hardin)

Column 1 of A284455 and of A307796.

a:= proc(n) option remember; `if`(n<6, [0$3, 1, 2, 6][n+1],
      ((n+2)*(5*n^4-98*n^3+661*n^2-1680*n+1164) *a(n-1)
       -4*(2*n^5-37*n^4+226*n^3-442*n^2-87*n+204) *a(n-2)
       -2*(3*n^4-63*n^3+376*n^2-468*n+264) *a(n-3)
       +2*(8*n^5-155*n^4+1060*n^3-3035*n^2+3738*n-1752) *a(n-4)
       -4*(5*n^5-101*n^4+750*n^3-2450*n^2+3312*n-1248) *a(n-5)
       +4*(2*n-9)*(n^4-16*n^3+85*n^2-150*n+48) *a(n-6)) /
       ((n+3)*(n^4-20*n^3+139*n^2-372*n+300)))
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Apr 23 2018

Recursion: see Maple program. - Alois P. Heinz, Apr 23 2018

A286209 Number of n X 1 0..1 arrays with the number of 1's king-move adjacent to some 0 two less than the number of 0's adjacent to some 1.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 3, 10, 24, 60, 134, 304, 656, 1420, 2996, 6312, 13112, 27167, 55825, 114412, 233282, 474563, 962159, 1947098, 3931288, 7925708, 15952866, 32072580, 64404708, 129213082, 259009006, 518818124, 1038549912, 2077775396, 4154785904, 8304424080

Offset: 0

R. H. Hardin, May 04 2017

nonn

			All solutions for n=7
..0. .0. .0
..1. .0. .1
..0. .1. .0
..0. .0. .0
..0. .0. .1
..1. .1. .0
..0. .0. .0

Alois P. Heinz, Table of n, a(n) for n = 0..3327 (terms n = 1..210 from R. H. Hardin)

Column k=1 of A286216.

Column k=2 of A307796.

b:= proc(n, t, h, c) option remember; `if`(abs(c-2)>n, 0, `if`(n=0, 1,
      b(n-1, [1, 3, 1][t], 2, c-`if`(h=3, 1, 0))+
      b(n-1, 2, [1, 3, 1][h], c+`if`(t=3, 1, 0))))
    end:
a:= n-> b(n, 1$2, 0):
seq(a(n), n=0..40);  # Alois P. Heinz, Apr 29 2019

b[n_, t_, h_, c_] := b[n, t, h, c] = If[Abs[c - 2] > n, 0, If[n == 0, 1,
     b[n - 1, {1, 3, 1}[[t]], 2, c - If[h == 3, 1, 0]] +
     b[n - 1, 2, {1, 3, 1}[[h]], c + If[t == 3, 1, 0]]]];
a[n_] := b[n, 1, 1, 0];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Jun 27 2022, after Alois P. Heinz *)

cp's OEIS Frontend

A303696 Number A(n,k) of binary words of length n with k times as many occurrences of subword 101 as occurrences of subword 010; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A164146 Number of binary strings of length n with equal numbers of 010 and 101 substrings.

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A284449 Number of n X 1 0..1 arrays with the number of 1's king-move adjacent to some 0 one less than the number of 0's adjacent to some 1.

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A286209 Number of n X 1 0..1 arrays with the number of 1's king-move adjacent to some 0 two less than the number of 0's adjacent to some 1.

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