A097230 Triangle read by rows: number of binary sequences with no isolated 1's.
1, 1, 0, 1, 0, 1, 1, 0, 2, 1, 1, 0, 3, 2, 1, 1, 0, 4, 3, 3, 1, 1, 0, 5, 4, 6, 4, 1, 1, 0, 6, 5, 10, 9, 5, 1, 1, 0, 7, 6, 15, 16, 13, 6, 1, 1, 0, 8, 7, 21, 25, 26, 18, 7, 1, 1, 0, 9, 8, 28, 36, 45, 40, 24, 8, 1, 1, 0, 10, 9, 36, 49, 71, 75, 59, 31, 9, 1, 1, 0, 11, 10, 45, 64, 105, 126, 120, 84, 39, 10, 1
Offset: 0
Examples
T(6,4) = 6 counts 001111, 011011, 011110, 110011, 110110, 111100.
Table begins:
\ k 0, 1, 2,
n
0 | 1;
1 | 1, 0;
2 | 1, 0, 1;
3 | 1, 0, 2, 1;
4 | 1, 0, 3, 2, 1;
5 | 1, 0, 4, 3, 3, 1;
6 | 1, 0, 5, 4, 6, 4, 1;
7 | 1, 0, 6, 5, 10, 9, 5, 1;
8 | 1, 0, 7, 6, 15, 16, 13, 6, 1;
...
Links
- Alois P. Heinz, Rows n = 0..140, flattened
Crossrefs
Programs
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Maple
b:= proc(n, w, s) option remember; `if`(n=0, `if`(s in [1, 21], 0, x^w), `if`(s in [1, 21], 0, b(n-1, w, irem(s, 10)*10))+b(n-1, w+1, irem(s, 10)*10+1)) end: T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, 0, 22)): seq(T(n), n=0..14); # Alois P. Heinz, Mar 03 2020 -
Mathematica
a[n_, 0]/;n>=0 := 1; a[n_, k_]/;k>n || k<0 :=0; a[n_, 1]:=0; a[2, 2]=1; a[n_, k_]/;n>=3 && 2 <= k <= n := a[n, k] = 1 + Sum[a[n-(r+1), k-j], {r, 2, n-1}, {j, Max[2, r-1-(n-k)], Min[r, k]}] (* This recurrence counts a(n, k) by r = location of first 1 followed by a 0, j = length of run which this first 1 terminates. *)
Formula
G.f.: (1-x*y+x^2*y^2)/( (1-x)*(1-x*y) -x^3*y^2 ) = Sum_{n>=0, k>=0} T(n,k) x^n y^k.
From Alois P. Heinz, Mar 03 2020: (Start)
Sum_{k=1..n} k * T(n,k) = A259966(n).
Sum_{k=1..n} k^2 * T(n,k) = A332863(n). (End)
Comments