A145113 -id:A145113 - cp's OEIS frontend

A145111 Square array A(n,k) of numbers of length n binary words with fewer than k 0-digits between any pair of consecutive 1-digits (n,k >= 0), read by antidiagonals.

1, 1, 2, 1, 2, 3, 1, 2, 4, 4, 1, 2, 4, 7, 5, 1, 2, 4, 8, 11, 6, 1, 2, 4, 8, 15, 16, 7, 1, 2, 4, 8, 16, 27, 22, 8, 1, 2, 4, 8, 16, 31, 47, 29, 9, 1, 2, 4, 8, 16, 32, 59, 80, 37, 10, 1, 2, 4, 8, 16, 32, 63, 111, 134, 46, 11, 1, 2, 4, 8, 16, 32, 64, 123, 207, 222, 56, 12, 1, 2, 4, 8, 16, 32, 64, 127, 239, 384, 365, 67, 13

Offset: 0

Alois P. Heinz, Oct 02 2008

			A(4,1) = 11, because 11 binary words of length 4 have fewer than 1 0-digit between any pair of consecutive 1-digits: 0000, 0001, 0010, 0100, 1000, 0011, 0110, 1100, 0111, 1110, 1111.
Square array A(n,k) begins:
  1,  1,  1,  1,  1,  1, ...
  2,  2,  2,  2,  2,  2, ...
  3,  4,  4,  4,  4,  4, ...
  4,  7,  8,  8,  8,  8, ...
  5, 11, 15, 16, 16, 16, ...
  6, 16, 27, 31, 32, 32, ...

Alois P. Heinz, Antidiagonals n = 0..140

Columns 0-9 give: A000027(n+1), A000124, A000126(n+1), A007800(n+1), A145112, A145113, A145114, A145115, A145116, A145117.
Main diagonal gives A000079.
Cf. A141539.

f:= proc(n,k) option remember; local j; if n=0 then 1 elif n<=k then 2^(n-1) else add(f(n-j, k), j=1..k) fi end: g:= proc(n,k) option remember; if n<0 then 0 else g(n-1,k) +f(n,k) fi end: A:= (n,k)-> `if`(n=0, g(0,k), A(n-1,k) +g(n-1,k)): seq(seq(A(n, d-n), n=0..d), d=0..14);

a[n_, k_] := SeriesCoefficient[(1 - x + x^(k+1))/(1 - 3*x + 2*x^2 + x^(k+1) - x^(k+2)), {x, 0, n}]; a[0, ] = 1; Table[a[n-k, k], {n, 0, 14}, {k, n, 0, -1}] // Flatten (* _Jean-François Alcover, Jan 15 2014 *)

G.f. of column k: (1-x+x^(k+1))/(1-3*x+2*x^2+x^(k+1)-x^(k+2)).

A242235 Number of length n+4+1 0..4 arrays with every value 0..4 appearing at least once in every consecutive 4+2 elements, and new values 0..4 introduced in order.

Original entry on oeis.org

15, 35, 71, 139, 271, 531, 1047, 2059, 4047, 7955, 15639, 30747, 60447, 118835, 233623, 459291, 902943, 1775139, 3489831, 6860827, 13488031, 26516771, 52130599, 102486059, 201482287, 396103747, 778719463, 1530922155, 3009713711

Offset: 1

R. H. Hardin, May 08 2014

nonn

			Some solutions for n=5:
  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
  0  1  1  1  1  1  1  1  1  0  1  1  0  1  1  1
  1  2  2  2  2  0  2  0  0  1  2  2  1  2  2  2
  2  3  0  1  3  2  3  2  2  2  2  3  2  0  3  0
  3  2  3  3  4  3  0  3  3  3  3  4  3  3  0  3
  4  4  4  4  1  4  4  4  4  4  4  1  4  4  4  4
  1  0  1  0  0  1  1  0  0  2  0  0  0  0  1  2
  0  1  2  3  2  2  2  1  1  0  1  2  1  1  2  1
  4  3  3  2  2  0  3  1  3  1  2  4  2  2  3  0
  2  0  0  1  3  1  4  2  2  0  1  3  2  2  0  1

R. H. Hardin, Table of n, a(n) for n = 1..210

Cf. A004767, A145113, A242239.

Column 4 of A242239.

Empirical: a(n) = a(n-1) + a(n-2) + a(n-3) + a(n-4) + a(n-5).
Conjecture: a(n) = 16*A145113(n-1) + A004767(n-2), n > 1. - R. J. Mathar, Aug 16 2017
Empirical g.f.: x*(15 + 20*x + 21*x^2 + 18*x^3 + 11*x^4) / (1 - x - x^2 - x^3 - x^4 - x^5). - Colin Barker, Oct 31 2018

cp's OEIS Frontend

A145111 Square array A(n,k) of numbers of length n binary words with fewer than k 0-digits between any pair of consecutive 1-digits (n,k >= 0), read by antidiagonals.

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