A006979 -id:A006979 - cp's OEIS frontend

A006980 Compositions: 6th column of A048004.

1, 2, 5, 12, 28, 64, 143, 315, 687, 1485, 3186, 6792, 14401, 30391, 63872, 133751, 279177, 581040, 1206151, 2497895, 5161982, 10646564, 21919161, 45052841, 92461171, 189489255, 387830160, 792810956, 1618840800, 3301999647

Offset: 6

Simon Plouffe

a(n) is the number of binary strings of length n-1 whose longest run of 1s has length 5. - Félix Balado, May 20 2025

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
J. L. Yucas, Counting special sets of binary Lyndon words, Ars Combin., 31 (1991), 21-29.

Alois P. Heinz, Table of n, a(n) for n = 6..1000
J. L. Yucas, Counting special sets of binary Lyndon words, Ars Combin., 31 (1991), 21-29. (Annotated scanned copy)
Index entries for linear recurrences with constant coefficients, signature (2,1,0,-1,-2,-4,-5,-4,-3,-2,-1).

Cf. A048004, A006979, A000102, A000100, A000071.

a:= n-> (Matrix(11, (i,j)-> if i=j-1 then 1 elif j=1 then [2, 1, 0, -1, -2, -4, -5, -4, -3, -2, -1][i] else 0 fi)^n) [1,7]: seq(a(n), n=6..40); # Alois P. Heinz, Oct 29 2008

Vec(1/(1-x-x^2-x^3-x^4-x^5)/(1-x-x^2-x^3-x^4-x^5-x^6)+O(x^99)) \\ Charles R Greathouse IV, Jan 10 2013

G.f.: x^6 / ((1-x-x^2-x^3-x^4-x^5) * (1-x-x^2-x^3-x^4-x^5-x^6)). - Alois P. Heinz, Oct 29 2008

G.f.: x^6 * (1-x)^2 / ((1-2*x+x^6) * (1-2*x+x^7)). - Félix Balado, May 20 2025

Corrected definition: 6th column of A048004. - Geoffrey Critzer, Nov 09 2008

A048003 Triangular array T read by rows: T(h,k) = number of binary words of length h and maximal runlength k.

Original entry on oeis.org

2, 2, 2, 2, 4, 2, 2, 8, 4, 2, 2, 14, 10, 4, 2, 2, 24, 22, 10, 4, 2, 2, 40, 46, 24, 10, 4, 2, 2, 66, 94, 54, 24, 10, 4, 2, 2, 108, 188, 118, 56, 24, 10, 4, 2, 2, 176, 370, 254, 126, 56, 24, 10, 4, 2, 2, 286, 720, 538, 278, 128, 56, 24, 10, 4, 2, 2, 464, 1388, 1126, 606, 286, 128, 56, 24, 10, 4, 2

Offset: 1

Clark Kimberling

			Rows: {2}; {2,2}; {2,4,2}; {2,8,4,2}; ...
T(3,2) = 4, because there are 4 binary words of length 3 and maximal runlength 2: 001, 011, 100, 110. - _Alois P. Heinz_, Oct 29 2008

Alois P. Heinz, Rows n = 1..141, flattened

T(h,2) = 2*a(h+1) for h=2, 3, ..., where a=A000071.
T(h,3) = 2*b(h) for h=3, 4, ..., where b=A000100.
T(h,4) = 2*c(h) for h=4, 5, ..., where c=A000102.
Cf. A048004.
Columns 5, 6 give: 2*A006979, 2*A006980. Row sums give: A000079.
Cf. A229756.

gf:= proc(n) 2*x^n/ (1-add(x^i, i=1..n-1))/ (1-add(x^j, j=1..n)) end:
T:= (h,k)-> coeff(series(gf(k), x, h+1), x, h):
seq(seq(T(h,k), k=1..h), h=1..13);  # Alois P. Heinz, Oct 29 2008

gf[n_] := 2*x^n*(x^2-2*x+1) / (x^(2*n+1)-2*x^(n+2)-x^(n+1)+x^n+4*x^2-4*x+1); t[h_, k_] := Coefficient[ Series[ gf[k], {x, 0, h+1}], x, h]; Table[ Table[ t[h, k], {k, 1, h}], {h, 1, 13}] // Flatten (* Jean-François Alcover, Oct 07 2013, after Alois P. Heinz *)

G.f. of column k: 2*x^k / ((1-Sum_{i=1..k-1} x^i) * (1-Sum_{j=1..k} x^j)). - Alois P. Heinz, Oct 29 2008
T(n, k) = 0 if k < 1 or k > n, 2 if k = 1 or k = n, 2T(n-1, k) + T(n-1, k-1) - 2T(n-2, k-1) + T(n-k, k-1) - T(n-k-1, k) otherwise (cf. similar formula for A048004). This is a simplification of the L-shaped sum T(n-1, k) + ... + T(n-k, k) + ... + T(n-k,1). - Andrew Woods, Oct 11 2013
For n > 2k, T(n, n-k) = 2*A045623(k). - Andrew Woods, Oct 11 2013

More terms from Alois P. Heinz, Oct 29 2008

cp's OEIS Frontend

A006980 Compositions: 6th column of A048004.

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