Fibonacci

Compositions with 1 or 2 elements, F = 1 + X·F + 2(X)·F
A080599 Bob Proctor

1, 1, 3, 12, 66, 450, 3690, 35280, 385560,...

$$a_n = n.a_{n-1} + {{n(n-1)} \over {2}} a_{n-2}$$

the types are
 * N/A
 * 1
 * 1+1, 2
 * 1+1+1, 1+2, 2+1
 * 1+1+1+1, 1+1+2, 1+2+1, 2+1+1, 2+2

the native construction H = 1 + Y+ HX + HXY
A000045

A102426 triangle of Fibonacci parts

The number of binary strings of length n without consecutive Ys is the Fibonacci number Fn+2


 * 1

Let an be strings of length n that start with 0, and bn n strings that starts with 1.Then:


 * an = an-1 + bn-1
 * bn = an-1 ; by summing we have:
 * an + bn = (an-1 + bn-1) + (an-2 + bn-2)

so we got the Fibonacci recurrence : Fn = Fn-1 + Fn-2

In terms of species,
 * F = X·F + X·G + X and
 * G = Y·F + Y = Y·X·F + Y·X·G + Y·X + Y that gives
 * F+G = (X + Y·X)·(F+G+1) + Y

By passing to e.g.f. for 1 + F + G we get:
 * $$GF = e.g.f. = { 1+y \over 1-x-xy } $$

Compositions with no singles L(E–X–1)
A032032

1, 0, 1, 1, 7, 21, 141, 743, 5699, 42241,...

$$ e.g.f. = { 1 \over 2 + x - exp(x)} $$

1 + E·F= F + F + X·F


 * 0
 * 2
 * 3
 * 2.2, 4
 * 2.3, 3.2, 5
 * 2.2.2, 2.4, 3.3, 4.2, 6
 * 2.2.2, 2.4, 3.3, 4.2, 6

Odd compositions, F = 1 + Odd.F
A006154 : 1, 1, 2, 7, 32, 181, 1232, 9787, 88832,...

$$ e.g.f. = \frac 1 {1 - sinh(x)}$$


 * 0
 * 1
 * 1.1
 * 3, 1.1.1
 * 3.1, 1.3, 1.1.1.1
 * 5, 3.1.1, 1.3.1, 1.1.3, 1.1.1.1.1