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In honour of Fibonacci.

Studying an artificial intelligence approach to programming the other day —I read the most weird documents!— I was reminded of the Fibonacci sequence. given by

 Fi = 0 , F2 = 1 , Fn = Fn-1 + Fn-2 (-inf < n < +inf)
For N ≥ 2 the relation
 R: x = FN
is trivially established by the program
 y, x, i := 0, 1, 2 {Y = Fi-1 and x = Fi and 2 ≤ i ≤ N}; (1) do i ≠ N → y, x, i := x, x + y, i + l .od {R}
a program with a time-complexity proportional to N; I remembered —although I did not know the formulae— that R can also be established in a number of operations proportional to log(N) and wondered —as a matter of fact: I still wonder— how proponents of “program transformations” propose to transform the linear algorithm (1) into the logarithmic one.

Yesterday evening I was wondering whether I could reconstruct the logarithmic scheme for the Fibonacci sequence, and whether similar schemes existed for higher order recurrence relations (for a k ≥ 2 ):

 F1 = F2 = ... = Fk-1 = 0 , Fk = 1 Fn = Fn-1 + ... + Fn-k (-inf < n < +inf) (2)

Eventually I found a way of deriving these schemes. For k = 2 , the normal Fibonacci numbers, the method leads to the well-known formulae

 F2j = fj2 + Fj+12 F2j+1 = (2Fj + Fj+1) * Fj+1 or F2j-1 = (2fj+1 - Fj) * Fj .
This note is written, because I liked my general derivation. I shall describe it for k = 3 .
Because for k = 3 we have F1 = F2 = 0 and F3 = l we may write
 Fn = F3 * Fn + (F2 + F1)* Fn-1 + F2 * Fn-2 (3)
From (3) we deduce the truth of
 Fn = Fi+3* Fn-i + (Fi+2 + Fi+1)* Fn-i-1+ Fi+2* Fn-i-2 (4)
for i = 0 . The truth of (4) for all positive values of i is derived by mathematical induction; the induction step consists of
1)     substituting Fn-i-1 + Fn-i-2 + Fn-i-3 for Fn-i
2)     combining after rearrangement Fi+3 + Fi+2 + Fi+1 for Fi+4 .
(The proof for negative values of i is done by performing the induction step the other way round.) Substituting in (4) n = 2j and i = j-l we get
 F2j = Fj2 * Fj+1 + (Fj+1 + Fj)* Fj + Fj+1* Fj-1
and, by substituting Fj+2 - Fj+1 - Fj for Fj-1 , and subsequent rearranging
 F2j = Fj2 + (2Fj+2 - Fj+1)* Fj+1 (5)
Substituting in (4) n = 2j+l and i = j-l we get
 F2j+l = Fj+2 + (Fj+1 + Fj)* Fj+1 + Fj+1 * Fj = (2Fj + Fj+1) * Fj+1 + Fj+22 (6)
Formulae (5) and (6) were the ones I was after.

Note. For k = 4 the analogue to (4) is Fn = Fi+4 * Fn-i + (Fi+3 + Fi+2 + Fi+1)* Fn-i-1 + (Fi+3 + Fi+2)* Fn-i-2 + Fi+3 * Fn-i-3 · (End of note.)

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 Plataanstraat 5 prof.dr.Edsger W.Dijkstra 5671 AL Nuenen Burroughs Research Fellow
 The Netherlands

Transcribed by Martin P.M. van der Burgt
Last revision 2015-01-29 .