However, sometimes all we have is a recursive definition and we do not know any direct formula for the general term of some series.
Also, recursive definitions are often much easier to find than a direct formula and also lend themselves to a nice method of
proof that the recursive definition is indeed correct.
We can count the number of derangements on `n` people as follows:
Take the person labelled `1`. He can end up in any of `n1` seats. Suppose he ends up in seat `i`. Then
consider the person labelled `i`. For each of the `n1` possibilities for `i`,
the person labelled `i` now cannot sit in her original seat (number `1` is sitting there) and so can either end
up in seat `1` or one of the other `n2` seats.
#seats(people)  1  2  3  4  5  6  ... 

#derangements  0  1  2  9  44  265  ... 
# Fixed points  0  1  2  3  Total 

Perms  (2,3,1) (3,1,2)  (1,3,2) (3,2,1) (2,1,3)    (1,2,3)  6 
# Fixed points  0  1  2  3  4  Total=n! 

# Perms{1}  0  1  1  
# Perms {1,2}  1  0  1  2  
# Perms {1,2,3}  2  3  0  1  6  
# Perms {1,2,3,4}  9  8  6  0  1  24 
As used here  Vajda  Dunlap  Knuth  Definition  Description  

Phi Φ 
τ  τ  φ, α  `(sqrt5 + 1)/2 = 1.6180339...`  Koshy uses α (page 78)  
phi φ  σ  φ  β  `(sqrt51)/2= 0.6180339...`  Koshy uses β (page 78)  
abs(x) x  x  x  x  abs(x) = x if x≥0; abs(x) = x if x<0  the absolute value of a number, its magnitude; ignore the sign;  
floor(x) `__x__`;  [x]  trunc(x), not used for `x < 0`  `__x__`  the nearest integer ≤ x. 
When `x > 0`, this is "the integer part of x" or "truncate x"
i.e. delete any fractional part after the decimal point. 3=floor(3)=floor(3.1)=floor(3.9), 4=floor(4)=floor(3.1)=floor(3.9)  
round(x) `[x]` 

trunc(x + 1/2)  the nearest integer to x; trunc(x+0.5)  3=round(3)=round(3.1), 4=round(3.9), 4=round(4)=round(3.9), 3=round(3.1) 4=round(3.5), 3=round(3.5)  
ceil(x) `~x~`;      `~x~`;  the nearest integer ≥ x.  3=ceil(3), 4=ceil(3.1)=ceil(3.9), 3=ceil(3)=ceil(3.1)=ceil(3.9)  
fract(x) frac(x)      x mod 1  x  floor(x)  the fractional part of x, i.e. the part of abs(x) after the decimal point  





_{n}C_{r} n choose r; the element in row n column r of Pascal's Triangle the coefficient of x^(r) in (1+x)^(n) the number of ways of choosing r objects from a set of n different objects. n≥0 and r≥0 (otherwise value is 0) 
Fibonaccitype series with the rule S(i) = S(i−1) + S(i−2) for all integers i:
i ... 6 5 4 3 2 1 0 1 2 3 4 5 6 ... Fibonacci
F(i)... 8 5 3 2 1 1 0 1 1 2 3 5 8 ... Lucas
L(i)... 18 11 7 4 3 1 2 1 3 4 7 11 18 ... General Fib
G(a,b,i)... 13a8b 8a+5b 5a3b 3a+2b 2ab a+b a b a+b a+2b 2a+3b 3a+5b 5a+8b ...
Note that sequences may have many different types of recurrence that can be used to define them precisely. For example, the natural numbers 0,1,2,3,4,5,6,... are defined by each of the following
Linear homogenous recurrence relations with constant coefficients
...have the property that
P  Q  `s_0`  `s_1, s_2, ...`  name  Generating Fn  OEIS 

`1`  `0`  `k`  `k,k,k,k,k...`  Constant series of k's  `k/(1x)`  `1,1,1,1,1...`A000012 `2,2,2,2,2...`A007395 `3,3,3,3,3...`A010701 `4,4,4,4,4...`A010709 `5,5,5,5,5...`A010716 ... 
`1`  `0`  `1`  `1,1,1,1,1,...(1)^n,...`  The Alternating Series of 1's  `1/(1+x)`  A033999 
`1`  `d`  `a`  `a+d,a+2d,a+3d,...,a+n d,...`  Arithmetic Series  `(a  a x + d x)/(x1)^2`  1,2,3,4,...,n,... A000027 0,2,4,6,8,...,2n,... A005843 1,3,5,7,9,...,2n+1,... A005408 0,3,6,9,12,...,3n,... A008585 1,4,7,10,13,...,3n+1,... A016777 ... 
`r`  `0`  `a`  `a r, a r^2, a r^3,..., a r^n, ...`  Geometric Series  `a/(1r x)`  1,2,4,8,16,32,...2^(n) A000079 1,3,9,27,81,...3^(n)A000244 3,6,12,24,48,96,...3 2^(n) A007283 5,10,20,40,80,... 5 2^(n) A020714 2,6,18,54,162,... 2 3^(n) A008776 4,12,36,108,324,... 4 3^(n) A003946 
`P`  `Q`  `s_0`  `s_1`  `s_2, s_3, ...`  name  Generating Fn  OEIS 

1  0  1,1,2,3,5,8...  Fibonacci F(n1)  `(1x)/(1xx^2)`  A000045  
1  1  0  1  1,2,3,5,8...  Fibonacci F(n)  `x/(1xx^2)`  A000045 
2  1  3,4,7,11,18,...  Lucas L(n)  `(2x)/(1xx^2)`  A000032  
F(0)+F(k)  F(1)+F(k+1)  ...  F(n) + F(n+k)  `(F(k+1)+x(F(k1)+1))/(1xx^2)`  k=0: A006355, k=1:F(n+1) k=2: L(n+1), k=3: 2F(n+1) k=4: 3F(n+1), k=5: F(n)+L(n+1) k=6:  
a  b  a+b,a+2b,2a+3b,...  Generalised Fibonacci G(a,b,n) = F(n1) a + F(n) b 
`(a+(ba)x)/(1xx^2)`  G(a,b)  
1  1  0  1  1,2,3,5,8...  Fibonacci(  n )  `x/(1+xx^2)`  A039834 
1  0  1,1,2,3,5,8...  F( 1  n )  `(1+x)/(1+xx^2)`  
a  b  ab,a+2b,2a3b,3a+5b,...  F( 1  n ) a + F( n ) b  `(a+(a+b)x)/(1+xx^2)`  
1  2  0  1  1,3,5,11,21...  Jacobsthal J(n) (2^(n)  (1)^(n))/3 
`x/(1xx^2)`  A001045 
1  0  2,2,6,10,22,42...  2 J(n1) (2^(n)  (1)^(n))2/3 
`(1x)/(1xx^2)`  A078008  
2  1  5,7,17,31...  JacobsthalLucas 2^(n) + (1)^(n) 
`(2x)/(1xx^2)`  A014551  
a  b  2a+b,2a+3b,6a+5b,10a+11b,...  2a J(n1) + b J(n)  `(a+x(ba))/(1xx^2)`  
2  1  0  1  2,5,12,29...  Pell Denominators of CF convergents to √2 
`x/(12xx^2)`  A000129 
1  1  3,7,17...  Numerators of CF convergents to √2  `(1x)/(1xx^2)`  A001333  
1  0  1,2,5,12,29...  Pell(n1)  `(12x)/(12xx^2)`  A000129  
2  2  6,14,34...  Companion Pell, PellLucas  `(22x)/(12xx^2)`  A002203  
a  b  a+2b,2a+5b,5a+12b,..  Pell(n) a + Pell(n+1) b  `(1+(b2a)x)/(12xx^2)`  
2  1  0  1  2,3,4,5...  the whole numbers  `x/(12x+x^2)`  A000027 
1  0  1,2,3,4,5...  the negative numbers  `(x)/(12x+x^2)`  A001478  
0  2  4,6,8...  the even numbers  `(2x)/(12x+x^2)`  A005843  
0  3  6,9,12,15...  3n the sum of 3 consecutive integers 
`3/(12x+x^2)`  A008585  
1  3  5,7,9...  the odd numbers  `(1+x)/(12x+x^2)`  A005408  
2  6  10,14,18,22..  4n+2 Numbers not the side of any primitive Pythagorean triangle. The sum of 4 consecutive numbers. 
`(2(53x))/(12x+x^2)`  A016825  
a  b  2ba,3b2a,4b3a...  n b  (n1) a  `(a+(b2a)x)/(12x+x^2)`  
a  a+d  a+2d, a+3d,...,a + n d, ...  Arithmetic Series  `(ax(ad))/(x1)^2 = (aax+dx)/(12x+x^2)`  
3  1  0  1  3,10,33,109...  Ratios of terms are CF convergents to `(3+sqrt13)/2`  `x/(13xx^2)`  A006190 
1  0  1,3,10,33,109...  Ratios of terms are CF convergents to `(3+sqrt13)/2`  `(13x)/(13xx^2)`  A006190  
3  1  0  1  3,8,21,...  `F(2n) = F(n+1)^2  F(n1)^2` `= ( F(n+2)^2  F(n2)^2 )/3` `= ( F(n+3)^2  F(n3)^2 )/8 = ...` 
`x/(13x+x^2)`  A001906 
1  0  1,3,8,21,...  `F(2  2n ) =  F( 2 n  2 )`  `(13x)/(13x+x^2)`  (A001906)  
1  2  5,13,34,89...  `F(2n+1) = F(n)^2 + F(n+1)^2`  `(1x)/(13x+x^2)`  A001519  
5  10  25,65,170,...  L(n)^{2} + L(n+1)^{2} = 5 F(2n+1) = 5 F(n)^{2} + 5F(n+1)^{2} 
`(5(1x))/(13x+x^2)`  A106729  
2  3  7,18,47,123...  L(2n)  `(23x)/(13x+x^2)`  A005278  
1  4  11,29,76,199...  L(2n+1)  `(1+x)/(13x+x^2)`  A002878  
a  b  3ba,8b3a,21b8a...  F(2n) b − F(2n −2) a  `(a+(b3a)x)/(13x+x^2)`  
3  2  0  1  3,7,15,31,63...  `2^n1`  `x/(13x+2x^2)=x/((12x)(1x))`  A000225 
2  3  5,9,17,33,65...  `2^n+1`  `(23x)/(13x+2x^2)`  A000051  
a  b  3b2a,7b6a,15b14a...  `(2^n  1) b  (2^n  2) a`  `(a+(b3a)x)/(13x+2x^2)`  
4  1  0  1  4,17,72...  Denominators of CF convergents to √5 =Fib(3n)/2 
`(x)/(1  4 x  x^2)`  A001076 
1  2  9,38,161...  Numerators of CF convergents to √5 = Lucas(3n)/2= Fib(3n)/2+F(3n1) 
`(1  2 x)/(1  4 x  x^2)`  A001077  
F(0)=0  F(3)=2  8,34,144...  F(3n)  `(2 x)/(1  4 x  x^2)`  A014445  
F(1)=1  F(4)=3  13,55...  F(3n+1)  `(1  x)/(1  4 x  x^2)`  A033887  
F(2)=1  F(5)=5  21,89...  F(3n+2)  `(1 + x)/(1  4 x  x^2)`  A015448  
a  b  a+4b,4a+17b,17a+72b...  ( F(3n) a + F(3n+3) b )/2  `(a + (b  4a) x)/(1  4 x  x^2)`  
6  1  0  1  6,35,204...  Numbers n such that n^{2} is a Triangular number = Pell(2n)/2 
`( x )/(1  6 x + x^2)`  A001109 
0  2  12,70,408...  Pell(2n), the even Pell numbers  `( 2 x )/(1  6 x + x^2)`  A001542  
1  5  29,169,985...  Hypotenuse, h, of Pythagorean triangles
with consecutive legs (a,a+1,h) Ratios of terms are convergents to CF [5; 1,4] = 3 + 2 √2 Pell(2n+1), the odd Pell numbers 
`(1  x)/(1  6 x + x^2)`  A001653  
a  b  −a+6b,−6a+35b,−35a+204b...  `(a + (6a+b) x)/(1  6 x + x^2)`  
7  1  0  1  7,48,329...  F(4n)/3  `(x)/(1  7 x + x^2)`  A004187 
a  b  7ba,48b7a,...  `(F(4n) b  F(4n4) a )/(3)`  `(a + (b  7a) x)/(1  7 x + x^2)` 
P  Q  s_{0}  s_{1}  name  Gen. Fn. 

L(k)  (1)^{k}  F(i)  F(i+k)  F(kn+i);i=0..k1  `(F(i) + ( F(k+i)  F(i)L(k) ) x)/(1  L(k) x  (1)^k x^2)` 
L(i)  L(i+k)  L(kn+i);i=0..k1  `(L(i) + ( L(k+i)  L(i)L(k) ) x)/(1  L(k) x  (1)^k x^2)`  
a  b  `(a + ( b  a L(k) ) x)/(1  L(k) x  (1)^k x^2)`  
n  1  0  1 
Denominators of CF convergents of ` (n + \sqrt(n^2+4)) /2`
whose CF is [n; n,n,n,n,...] 
`( x )/(1  n x  x^2)` 
1  n  Numerators of CF convergents of CF [n; n,n,n,n,...] 
`( 1 )/(1  n x  x^2)`  
P  Q  a  b  General second order recurrence  `( a  (a P  b) x )/(1  P x  Q x^2)` 
P  Q  R  s_{0}  s_{1}  s_{2}  s_{3} ...  name  Gen. Fn.  OEIS 

0  1  1  0  0  1  0,1,1,1,2,2,3,4,5,7...  Padovan  `( x^2)/(1  x^2  x^3)`  A000931 
0  2  1  0  1  1  2,3,5,8,13...  Fibonacci F(n+3)=2F(n+1)+F(n) Vajda8 with m=3) 
`(x )/((1  x  x^2) )`  A000045 
1  1  1  0  0  1  1,2,4,7,13,24...  Tribonacci numbers  `( x^2)/(1  x  x^2  x^3)`  A000073 
2  0  1  0  1  1  2,3,5,8,13...  Fibonacci B&Q(2003) Identity 16 
`(x )/((1  x  x^2 ))`  A000045 
2  2  1  0  1  1  4,9,25,64...  F(n)^{2}  `( x  x^2)/(1  2 x  2 x^2 + x^3)` = `( x ( 1  x ))/(( 1 + x )( 1  3 x + x^2))`  A007598 
0  1  2  6,15,...  `F(n)F(n+1)=(F(n+2)^(2)F(n1)^(2))/4`  `( x )/(1  2 x  2 x^2 + x^3)`  A001654  
0  2  3  10,24,...  F(n)F(n+2)  `( 2 x  x^2)/(1  2 x  2 x^2 + x^3)`  A059929  
0  3  5  16,39,...  `F(n1)F(n+2)=F(n+1)^(2)F(n)^(2)`  `( 3 x  x^2)/(1  2 x  2 x^2 + x^3)`  A226205  
0  4  8  24,60,160,...  `F(n+3)^(2)  F(n)^2` = 4 F(n) F(n+1) 
`( 4 x )/(1  2 x  2 x^2 + x^3)`  A192883  
0  5  8  26,63,...  F(n2)F(n+2)  `( 5 x  2 x^2)/(1  2 x  2 x^2 + x^3)`  A192883  
3  8  25  63,168...  `F(n+5)^(2)F(n)^(2)`  `( 3 + 2x + 3x^2)/(1  2 x  2 x^2 + x^3)`  
4  1  9  16,49,121...  L(n)^{2}  `( 4  7 x  x^2)/(1  2 x  2 x^2 + x^3)`  A001254  
2  3  12  28,77,198...  L(n)L(n+1)  `( 2  x + 2 x^2)/(1  2 x  2 x^2 + x^3)`  A215602  
a  b  c  −a+2b+2c, −2a+3b+6c, −6a+10b+15c ...  −F(n)F(n+1)a +F(n+2)G(b,c,n+1)  `( a + ( b2a ) x  ( 2a+2bc ) x^2)/(1  2 x  2 x^2 + x^3)`  
0  a  b  2a+2b, 3a+6b, ...  F(n)G(a,b,n1)  `( a x + ( b  2 a )x^2)/(1  2 x  2 x^2 + x^3)`  
a^{2}  b^{2}  (a+b)^{2}  (a+2b)^{2}, (2a+3b)^{2} ...  G(a,b,n)^{2}  `( a^(2) + (b^(2)  2 a^(2)) x  (a  b)^(2) x^2)/(1  2 x  2 x^2 + x^3)`  
ab  b(a+b)  (a+b)(a+2b)  (a+2b)(2a+3b), ...  G(a,b,n) G(a,b,n+1) =F_{n1}F_{n}a^{2} + (F_{n}F_{n+1}+F_{n1}^{2})ab +F_{n}F_{n+1}b^{2} 
`( a b  b ( a  b) x + a (a  b) x^2)/(1  2 x  2 x^2 + x^3)`  
3  3  1  0  1  2  3,4,5...  Natural Numbers  `( x )/((x  1)^(2))`  A000017 
0  1  3  6,10,15...  Triangular Numbers = n ( n + 1 ) / 2  `( x)/((x  1)^(3))`  A000217  
0  2  6  12,20,30...  Pronic Numbers = n ( n + 1 )  `( 2x)/((x  1)^(3))`  A002378  
0  1  4  9,16,25...  Square Numbers = n^{2}  `( x (x + 1))/((x  1)^(3))`  A000290  
0  1  5  12,22,35...  Pentagonal Numbers =n ( 3 n  1) / 2 
`( x (2 x + 1))/((x  1)^(3))`  A000326  
0  1  6  15,28,45...  Hexagonal Numbers =n ( 2 n  1) / 2 
`( x (3 x + 1))/((x  1)^(3))`  A000384  
0  1  N  3N3,6N8,10N15,...  Polygonal Numbers Figurate Numbers with N sides= n [ N (n1) /2  (n2) ] 
`( x + (N3) x^2)/((x  1)^(3))`  
1  4  10  19, 31, 46, ...  Centred Triangular Numbers =½` (3 n^(2)  3 n + 2)` `= 1 + 3 n(n1)/2` 
`( x (1 + x + x^2))/((x  1)^(3))`  A005448  
1  5  13  25,41,61, ...  Centred Square numbers = Hypotenuses h of Pythagorean triangles with sides (a, h1, h) = 4 Triangular(n) + 1 = `2 n^(2) + 2 n + 1` 
`( (1 + x)^( 2))/((x  1)^(3))`  A001844  
1  6  16  31, 51, 76, ...  Centred Pentagonal Numbers = ½ `(5 n^(2)  5 n + 2)` 
`( 1 + 3 x + x^2)/((x  1)^(3))`  A005891  
1  7  19  37, 61, 91, ...  Centred Hexagonal Numbers = ½ `(6 n^(2)  6 n + 2)` = 1  3n + 3n^{2} 
`( 1 + 4 x + x^2)/((x  1)^(3))`  A003215  
1  1+k  1+3k  1+6k, 1+10k, 1+15k, ...  Centred Polygonal(kgonal) Numbers = ½ `(k n^(2)  k n + 2)` 
`( 1 + (k  2) x + x^2)/((x  1)^(3))`  
7  7  1  0  3  20  119,696,...  Smallest leg,a, of Pythagorean triangles (a,a+1,h) 
`( 3  x)/(1  7 x + 7 x^2  x^3)`  A001652 
1  8  49  288,1681,...  The rank of a triangle number which is also square  `( 1 + x)/((1  x )( 1  6 x^2 + x^2))`  A001108  
8  8  1  0  1  9  64,441...  F(2n)^{2}  `( x + x^2)/(1  8 x + 8 x^2  x^3)` `=( x ( 1 + x ))/(( 1  x )( 1  7 x + x^2 ))`  A049684 
1^{2}  2^{2}  5^{2}  169...  F(2n+1)^{2}  `(1  4 x + x^2)/(1  8 x + 8 x^2  x^3)`  A081068  
15  15  1  0  1  12  165, 2296, ...  Rank of Pentagonal Numbers that are Triangular  `( x (1  3x))/((1  x)(1 14x+x^2))`  A046174 
0  1  20  285, 3976, ...  Rank of Triangular Numbers that are Pentagonal  `( x (1 + 5x))/((1  x)(1 14x+x^2))`  A046175  
17  17  1  0  2^{2}  8^{2}  1156...  F(3n)^{2}  `( 4 x  4 x^2)/( 1 + 17 x + 17 x^2  x^3)` `= ( 4 (x  1) x)/(( 1 + x )( 1  18 x + x^2 ))` 
A014729 
1^{2}  3^{2}  13^{2}  3025...  F(3n+1)^{2}  `(  1 + 8 x + x^2)/( 1 + 17 x + 17 x^2  x^3)`  
1^{2}  5^{2}  21^{2}  7921...  F(3n+2)^{2}  `(  1  8 x + x^2)/( 1 + 17 x + 17 x^2  x^3)`  
35  35  1  1  36  1225  41616, ...  Square Triangular numbers  `( 1 + x )/((1  x )( 1  34 x + x^2))`  A001110 
195  195  1  0  1  210  40755, ...  Pentagonal Triangular numbers  `(x( 1 + 15x) )/((x  1 )( 1  194 x + x^2))`  A001110 
9603  9603  1  1  9801  94109401  903638458801, ...  Square Pentagonal numbers  `( 1 + 198 x + x^2)/((1  x )( 1  9602 x + x^2))`  A036353 
P  Q  R  s_{0}  s_{1}  s_{2}  series  Gen.Fn.  

L(2k)+(1)^{k}  L(2k)(1)^{k}  (1)^{k}  F(i)^{2} = a  F(k+i)^{2} = b  F(2k+i)^{2} = c 
F(kn+i)^{2}  `( a  ( a P  b) x  ( a Q + b P  c ) x^2)/((x + (1)^(k))(x^2  L(2k) x + 1))`  for i in {0..k1}  
P  Q  R  a  b  c  General Third order Recurrence  `( a  ( a P  b) x  ( a Q + b P  c ) x^2)/(1  P x  Q x^2  R x^3)` 
P  Q  R  S  s_{0}  s_{1}  s_{2}  s_{3}  series  name  Gen. Fn.  OEIS 

3  6  3  1  0  1  1  8  8,27,125...  F(n)^{3}  `( 1  2x  x^2 )/(1  3 x  6 x^2 + 3 x^3 + x^4)` = `( 1  2x  x^2 )/(( x^2  x  1 )( x^2 + 4 x  1 ))`  A056570 
0  0  0  1  3,15,60,260  F(n1)F(n)F(n+1)/2  `( 1 )/(1  3 x  6 x^2 + 3 x^3 + x^4)`  A001655  
0  0  0  2  6,30,120,520  F(n1)F(n)F(n+1)  `( 2 )/(1  3 x  6 x^2 + 3 x^3 + x^4)`  A065563  
0  0  1  2  12,45,200...  F(n)^{2}F(n+1)  `( x  x^2 )/(1  3 x  6 x^2 + 3 x^3 + x^4)`  A066258  
0  0  3  10  48,195,840...  F(n1)F(n)F(n+2)  `( 3 x + x^2 )/(1  3 x  6 x^2 + 3 x^3 + x^4)`  A220362  
0  2  0  10  24,130,504...  F(n2)F(n)F(n+2)  `( 2  6 x  2 x^2 )/(1  3 x  6 x^2 + 3 x^3 + x^4)`  A226958  
1  0  6  15  80,312,1365...  F(n2)F(n)F(n+1)  `( x  3 x^2 )/(1  3 x  6 x^2 + 3 x^3 + x^4)`  A220360  
0  3  5  0  39,105,544  F(n2)F(n)F(n+3)  `( 3  9 x  2 x^2 )/(1  3 x  6 x^2 + 3 x^3 + x^4)`  
1  1  2  9  35,152...  F(n)^{3}+F(n+1)^{3}  `(1  x  3 x^2  3 x^3 )/(1  3 x  6 x^2 + 3 x^3 + x^4)`  A110224  
1  9  28  133  539,2322..  F(n)^{3}+F(n+2)^{3}  `(1 + 6 x  5 x^2  2 x^3 )/(1  3 x  6 x^2 + 3 x^3 + x^4)`  A226976  
F(n)^{3}+F(n+k)^{3}  `((1  2x  x^2)(1 + x^k))/(1  3 x  6 x^2 + 3 x^3 + x^4)`  
F(n)F(n+a)F(n+b)  `( (1)^(b)( F(a+1)F(b+1) x + G(L(b),F(b2),a) x^2  F(b1)F(a1) x^3 ))/(1  3 x  6 x^2 + 3 x^3 + x^4)` 
G(x) =  ∞ Σ i=0  `S(i) x^i`  `= S(0) + S(1) x + S(2) x^2 + S(3) x^3 + ...` 
To shift to the right (insert a 0 at the start of the series so all other terms have an index increased by 1),
multiply the GF by x; to shift to the left, divide by x.
There is much more on GFs on my Fibonomials page.
Fibonacci(n) 0,1,1,2,3,...  `(x)/(1  x  x^2)`  Lucas(n) 2,1,3,4,7,...  `(2  x)/(1  x  x^2)`  G(a,b,n) a,b,a+b,a+2b,...  `(a + (b  a) x)/( 1  x  x^2 )`  
Fibonacci(2n) 0,1,3,8,21,...  `(x)/( x^2  3 x + 1 )`  Lucas(2n) 2,3,7,18,...  `(2  3 x)/( x^2  3 x + 1 )`  G(a,b,2n) a,a+b,2a+3b,...  `(a + (b  2a) x)/( x^2  3 x + 1 )`  
Fibonacci(2n+1) 1,2,5,13,...  `(1  x)/( x^2  3 x + 1 )`  Lucas(2n+1) 1,4,11,29,...  `(1 + x)/( x^2  3 x + 1 )`  G(a,b,2n+1) b,a+2b,3a+5b,...  `(b + (a  b) x)/( x^2  3 x + 1 )`  
Fibonacci(3n) 2,8,34,144,...  `(2 x)/( 1  4 x  x^2 )`  Lucas(3n) 2,4,18,76,...  `(2  4 x)/( 1  4 x  x^2 )`  G(a,b,3n) a,a+2b,5a+8b,...  `(a + (2 b  3 a) x)/( 1  4 x  x^2 )`  
Fibonacci(3n+1) 1,3,13,55,...  `(3 + x)/( 1  4 x  x^2 )`  Lucas(3n+1) 3,11,47,199,...  `(3 x + 1)/( 1  4 x  x^2 )`  G(a,b,3n+1) a+b,3a+5b,13a+21b,...  `(b + (2 a  b) x )/( 1  4 x  x^2 )`  
Fibonacci(3n+2) 1,5,21,89,...  `(x + 1)/( 1  4 x  x^2 )`  Lucas(3n+2) 2,4,18,76,...  `(3  x)/( 1  4 x  x^2 )`  G(a,b,3n+2) a,a+2b,5a+8b,...  `(a + b + (b  a) x )/( 1  4 x  x^2)`  
Fibonacci(k n)  `(F(k) x)/(1  L(k) x + (1)^k x^2)`  Lucas(k n)  `(2  L(k) x)/(1  L(k) x + (1)^k x^2)`  G(a,b,kn)  `(a+(F(k)bF(k+1)a)x)/(1L(k)x+(1)^k x^2)`  
Fibonacci(n)^{2} `0^(2),1^(2),1^(2),2^(2),3^(2),...`  `(x  x^2 )/(1  2 x  2 x^2 + x^3)`  `Lucas(n)^(2)` `2^(2),1^(2),3^(2),4^(2),...`  `(4  7 x  x^2)/(1  2 x  2 x^2 + x^3 )`  `G(a,b,n)^(2)` `a^(2),b^(2),(a+b)^(2),... `  `(a^(2)+(b^(2)2a^(2))x(ab)^(2)x^2)/(12x2x^2+x^3)`  
Fib(n)Fib(n+1) 1×1,1×2,2×3,3×5,...  `( x)/( 1  2 x  2 x^2 + x^3)`  Lucas(n)Lucas(n+1) 2×1,1×3,3×4,4×7,...  `(2  x + 2 x^2)/(1  2 x  2 x^2 + x^3 )`  G(a,b,n)G(a,b,n+1) ab,b(a+b), (a+b)(a+2b),...  `(ab + b(ba)x + a(ab)x^2)/(12x2x^2+x^3)`  
Fibonacci(n)^{3} `0^(3),1^(3),1^(3),2^(3),3^(3),...`  `(x2x^2x^3)/(13x6x^2+3x^3+x^4)`  `Lucas(n)^(3)` `2^(3),1^(3),3^(3),4^(3),...`  `(823x24x^2+x^3)/(13x6x^2+3x^3+x^4)` 
The GF of `1,2,5,13,...` is that of
`Fib(2n+1)`
which is
`(1  x)/(x^2  3 x + 1)`>
so `1,0,2,0,5,0,13,...` has GF
`(1  x^2)/(x^4  3 x^2 + 1)`
To insert an extra 0 at the start, multiply the GF by x.
So the GF for the oddindexed Fibonacci numbers only in their correct positions in the Fibonacci series
with `Fib(2n+1)` is the coefficient of `x^(2n+1)`
is therefore `x (1  x^2)/(x^4  3 x^2 + 1)`
for the series 0,1,0,2,0,5,0,13,... .
Adding these two GFs, that is, for Fib(2n) as the coefficient of `x^(2n)`
and Fib(2n+1) as the coefficient of `x^(2n+1)`
should then give the complete
Fibonacci series GF:
0,0,1,0,3,0,8, 0,21, ... +We can check that `x^2/(x^4  3 x^2 + 1) + x (1  x^2)/(x^4  3 x^2 + 1) = x/(1  x  x^2)`
0,1,0,2,0,5,0,13, 0, ...
0,1,1,2,3,5,8,13,21,...
Multiplying a GF by a constant k multiples all the members of the series by k.
A series formed by adding two series S and T elementwise to form the series {S(n)+T(n) for n=1,2,3,...},
has a GF which is the sum of the two separate GFs for example for `a Fib_(n1) + b Fib_n = G(a,b)`.

Exponential Generating Functions For Fibonacci Identities C Church and Marjorie Bicknell, Fib Q vol 11 (1983) 275281  

Exponential Generating Functions For Fibonacci Identities C Church and Marjorie Bicknell, Fib Q vol 11 (1983) 275281 
Fibonacci and Phi in the Arts 
Fibonacci Home Page
WHERE TO NOW??? 
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