Leonardo numbers

The   Leonardo numbers   are a sequence of numbers defined by:

Leonardo numbers is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

   L(0) = 1
   L(1) = 1
   L(n) = L(n-1)  +    L(n-2)   +  1
       also
   L(n) =      2  *  Fib(n+1)   -  1

  where the   + 1   (herein)   will be known as the   add   number.

  where the   FIB   is the   Fibonacci numberss.

The task will be using the 3^rd equation (above) to calculate the Leonardo numbers.

Edsger W. Sijkstra   used them as an integral part of his   smoothsort   algorithm.

The first few Leonardo numbers are:

    1   1   3   5   9   15   25   41   67   109   177   287   465   753   1219   1973   3193   5167   8361  ··· 

Task

•   show the 1^st   25   Leonardo numbers, starting at L(0).
•   allow the first two Leonardo numbers to be specified   [for L(0) and L(1)].
•   allow the   add   number to be specified   (1 is the default).
•   show the 1^st   25   Leonardo numbers, specifying 0 and 1 for L(0) and L(1), and 0 for the add number.

(The last task requirement will produce the Fibonacci numbers.)

Show all output here.

Related tasks

•   Fibonacci number
•   Fibonacci n-step number sequences

See also

•   Wikipedia, Leonardo numbers
•   Wikipedia, Fibonacci numbers
•   OEIS Leonardo numbers

REXX

<lang rexx>/*REXX pgm computes Leonardo numbers, allowing the specification of L(0), L(1), and ADD#*/ numeric digits 500 /*just in case the user gets ka-razy. */ @.=1 /*define the default for the @. array.*/ parse arg N L0 L1 a# . /*obtain optional arguments from the CL*/ if N == | N =="," then N= 25 /*Not specified? Then use the default.*/ if L0\== & L0\=="," then @.0= L0 /*Was " " " " value. */ if L1\== & L1\=="," then @.1= L1 /* " " " " " " */ if a#\== & a#\=="," then @.a= a# /* " " " " " " */ say 'The first ' N " Leonardo numbers are:" /*display a title for the output series*/ if @.0\==1 | @.1\==1 then say 'using ' @.0 " for L(0)" if @.0\==1 | @.1\==1 then say 'using ' @.1 " for L(1)" if @.a\==1 then say 'using ' @.a " for the add number" say /*display blank line before the output.*/ $= /*initialize the output line to "null".*/

            do j=0  for N                       /*construct a list of Leonardo numbers.*/
            if j<2  then z=@.j                  /*for the 1st two numbers, use the fiat*/
                    else do                     /*··· otherwise, compute the Leonardo #*/
                         _=@.0                  /*save the old primary Leonardo number.*/
                         @.0=@.1                /*store the new primary number in old. */
                         @.1=@.0  +  _  +  @.a  /*compute the next Leonardo number.    */
                         z=@.1                  /*store the next Leonardo number in Z. */
                         end                    /* [↑]  only 2 Leonardo #s are stored. */
            $=$ z                               /*append the just computed # to $ list.*/
            end   /*j*/                         /* [↓]  elide the leading blank in  $. */

say strip($) /*stick a fork in it, we're all done. */</lang>

output   when using the default input:

The first  25  Leonardo numbers are:

1 1 3 5 9 15 25 41 67 109 177 287 465 753 1219 1973 3193 5167 8361 13529 21891 35421 57313 92735 150049

output   when using the input of:     12   0   1   0

The first  25  Leonardo numbers are:
using  0  for L(0)
using  1  for L(1)
using  0  for the  add  number

0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765 10946 17711 28657 46368