Fibonacci numbers are numbers in the Fibonacci sequence . We define the Fibonacci sequence recursively as and .

In Maxima we can compute the *n*-th Fibonacci number with the `fib`

function. The beginning of the sequence is

(%i1) makelist(fib(i), i, 1, 10); (%o1) [1,1,2,3,5,8,13,21,34,55]

We can compute the well-known explicit formula for

(%i2) load(solve_rec)$ (%i3) solve_rec(f[n]=f[n-1]+f[n-2], f[n], f[1]=1, f[2]=2); (%o3) f[n]=((sqrt(5)+1)^n*(sqrt(5)+5)*2^(-n-1))/5- ((sqrt(5)-5)*(sqrt(5)-1)^n*2^(-n-1)*(-1)^n)/5

or nicer

In this post I want to show how to use the explicit formula to investigate some interesting properties of the Fibonacci numbers. Usually we prove identities with Fibonacci numbers using induction, but here I also show how to find them.

### Proving identities

First we need to express using the golden ratio . We will use the `fibtophi`

function.

(%i4) fib(n)$ (%i5) fibtophi(%); (%o5) (%phi^n-(1-%phi)^n)/(2*%phi-1)

which gives . Maxima uses the identity to simplify expressions which contain .

A simple example is to “prove” the definition of Fibonacci numbers.

(%i6) fib(n) - fib(n-1) - fib(n-2); (%o6) fib(n)-fib(n-1)-fib(n-2) (%i7) ratsimp(fibtophi(%)); (%o7) 0

A little more interesting is the identity . I will use the latest `simplify_sum`

package from cvs.

(%i8) load(simplify_sum)$ (%i9) sum(k*fib(k), k, 1, n) - n*fib(n+2) + fib(n+3) - 2$ (%i10) fibtophi(simplify_sum(%))$ (%i11) ratsimp(%); (%o11) 0

The last example is from Wikipedia. It is the “divisibility by 11” property:

(%i12) sum(fib(n+k), k, 0, 9)/fib(n+6)$ (%i13) fibtophi(%)$ (%i14) ratsimp(%); (%o14) 11

### Finding identities

##### Division identities

We can find more identities like the divisibility property given above.

(%i15) fib_ratio(a,b) := ratsimp( fibtophi(sum(fib(n+i), i, 0, a)/fib(n+b)))$ (%i16) sublist( create_list([a,b,fib_ratio(a,b)], a, 1, 20, b, 1, a), lambda([l], is(integerp(last(l))))); (%o16) [[2,2,2],[5,4,4],[9,6,11],[13,8,29],[17,10,76]]

The last triple gives the identity

.

As a comparison, here is some similar code in Mathematica

In[1]:= fibRatio[a_, b_] := FullSimplify[ Sum[Fibonacci[n + i], {i, 0, a}]/Fibonacci[n + b]] In[2]:= Select[ Flatten[Table[{a, b, fibRatio[a, b]}, {a, 1, 10}, {b, 1, a}], 1], IntegerQ[#[[3]]] & ] Out[2]= {{2, 2, 2}, {9, 6, 11}}

Note that the Mathematica code (in Mathematica 7.0.0) runs much longer than the Maxima code, even though I searched in a smaller sample of ratios. It also misses the case [5,4,4].

##### Sums of Fibonacci numbers

Let’s find a simple expression for . We search for an identity of the form

where , , and are unknowns.

First we setup the expression for the identity.

(%i17) sm: sum(k^2*fib(k), k, 1, n) = sum(a[k-n]*n^2*fib(k), k, n+1, n+3) + sum(b[k-n]*n*fib(k), k, n+1, n+3) + sum(c[k-n]*fib(k), k, n+1, n+3) + d$ (%i18) sm1: simplify_sum(sm)$ (%i19) eq: fibtophi(sm1)$

Now we evaluate `eq`

so that we get 10 equations with 10 unknowns.

(%i20) makelist(''eq, n, 10, 19)$

Now solve the system. We get many solutions and pick one by setting all parameters in the solution to 0.

(%i21) sol: solve(%); solve: dependent equations eliminated: (8 10 9) (%o21) [[d=-8,c[3]=3-%r3,b[3]=-%r2-2,a[3]=-%r1,c[2]=%r3+2,b[2]=%r2, a[2]=%r1+1,c[1]=%r3,b[1]=%r2,a[1]=%r1]] (%i22) sol1: subst( map(lambda([x], x=0), %rnum_list), sol); (%o22) [[d=-8,c[3]=3,b[3]=-2,a[3]=0,c[2]=2,b[2]=0,a[2]=1,c[1]=0,b[1]=0,a[1]=0]]

Let’s look at the identity.

(%i23) subst(sol1[1], sm); (%o23) sum(k^2*fib(k),k,1,n)=-2*n*fib(n+3)+3*fib(n+3)+n^2*fib(n+2)+2*fib(n+2)-8 (%i24) map(factorsum, %); (%o24) sum(k^2*fib(k),k,1,n)=-(2*n-3)*fib(n+3)+(n^2+2)*fib(n+2)-8

which is

Of course this is not a proof yet. But the proof is easy:

(%i25) ratsimp(fibtophi(simplify_sum(rhs(%) - lhs(%)))); (%o25) 0