Metallic ratios

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Metallic ratios 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.

Many people have heard of the Golden ratio, phi (φ). Phi is just one of a series of related ratios that are referred to as the "Metallic ratios".

The Golden ratio was discovered and named by ancient civilizations as it was thought to be the most pure and beautiful (like Gold). The Silver ratio was was also known to the early Greeks, though was not named so until later as a nod to the Golden ratio to which it is closely related. The series has been extended to encompass all of the related ratios and was given the general name Metallic ratios (or Metallic means). Somewhat incongruously as the original Golden ratio referred to the adjective "golden" rather than the metal "gold".

Metallic ratios are the real roots of the general form equation: 

   x² - bx - 1 = 0

where the integer b determines which specific one it is.

Using the quadratic equation: 

  ( -b ± √(b² - 4ac) ) / 2a = x

Substitute in (from the top equation) 1 for a, -1 for c, and recognising that -b is negated we get: 

  ( b ± √(b² + 4) ) ) / 2 = x

We only want the real root: 

  ( b + √(b² + 4) ) ) / 2 = x

When we set b to 1, we get an irrational number: the Golden ratio. 

   ( 1 + √(1² + 4) ) / 2  =  (1 + √5) / 2 = ~1.618033989...

With b set to 2, we get a different irrational number: the Silver ratio. 

   ( 2 + √(2² + 4) ) / 2  =  (2 + √8) / 2 = ~2.414213562...

When the ratio b is 3, it is commonly referred to as the Bronze ratio, 4 and 5 are sometimes called the Copper and Nickel ratios, though they aren't as standard. After that there isn't really any attempt at standardized names. They are given names here on this page, but consider the names fanciful rather than canonical.

Note that technically, b can be 0 for a "smaller" ratio than the Golden ratio. We will refer to it here as the Platinum ratio, though it is kind-of a degenerate case.

Metallic ratios where b > 0 are also defined by the irrational continued fractions: 

   [b;b,b,b,b,b,b,b,b,b,b,b,b,b,b,b,b...]


So, The first 10 Metallic ratios are:

Metallic ratios
Name b Equation Value Continued fraction OEIS link
Platinum 0 (0 + √4) / 2 1 - -
Golden 1 (1 + √5) / 2 1.618033988749895... [1;1,1,1,1,1,1,1,1,1,1...] OEIS:A001622
Silver 2 (2 + √8) / 2 2.414213562373095... [2;2,2,2,2,2,2,2,2,2,2...] OEIS:A014176
Bronze 3 (3 + √13) / 2 3.302775637731995... [3;3,3,3,3,3,3,3,3,3,3...] OEIS:A098316
Copper 4 (4 + √20) / 2 4.23606797749979... [4;4,4,4,4,4,4,4,4,4,4...] OEIS:A098317
Nickel 5 (5 + √29) / 2 5.192582403567252... [5;5,5,5,5,5,5,5,5,5,5...] OEIS:A098318
Aluminum 6 (6 + √40) / 2 6.16227766016838... [6;6,6,6,6,6,6,6,6,6,6...] OEIS:A176398
Iron 7 (7 + √53) / 2 7.140054944640259... [7;7,7,7,7,7,7,7,7,7,7...] OEIS:A176439
Tin 8 (8 + √68) / 2 8.123105625617661... [8;8,8,8,8,8,8,8,8,8,8...] OEIS:A176458
Lead 9 (9 + √85) / 2 9.109772228646444... [9;9,9,9,9,9,9,9,9,9,9...] OEIS:A176522


There are other ways to find the Metallic ratios; one, (the focus of this task) is through successive approximations of Lucas sequences.

A Lucas sequence has the form xₙ = b * xₙ₋₁ + c * xₙ₋₂. The metallic ratios use Lucas sequences where c = 1 and where the first two terms are always 1, 1.

So for the Lucas sequence when b = 1: 

   xₙ = 1 * xₙ₋₁ + xₙ₋₂.

   1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144...

more commonly known as the Fibonacci sequence.

When b = 2: 

   xₙ = 2 * xₙ₋₁ + xₙ₋₂.

   1, 1, 3, 7, 17, 41, 99, 239, 577, 1393...


And so on.


To find the ratio by successive approximations, divide the (n+1)th term by the nth. As n grows larger, the ratio will approach the b metallic ratio.

For b = 1 (Fibonacci sequence): 

   1/1   = 1
   2/1   = 2
   3/2   = 1.5
   5/3   = 1.666667
   8/5   = 1.6
   13/8  = 1.625
   21/13 = 1.615385
   34/21 = 1.619048
   55/34 = 1.617647
   89/55 = 1.618182
   etc.

It converges, but pretty slowly. In fact, the Golden ratio has the slowest possible convergence for any irrational number.


Task

For each of the first 10 Metallic ratios; b = 0 through 9:

  • Generate the corresponding Lucas sequence.
  • Show here, on this page, at least the first 15 elements of the Lucas sequence.
  • Using successive approximations, calculate the value of the ratio accurate to 32 decimal places.
  • Show the value of the approximation at the required accuracy.
  • Show the value of n when the approximation reaches the required accuracy (How many iterations did it take?).

Optional, stretch goal - Show the value and number of iterations n, to approximate the Golden ratio to 256 decimal places.

You may assume that the approximation has been reached when the next iteration does not cause the value (to the desired places) to change.


See also


Factor

<lang factor>USING: combinators decimals formatting generalizations io kernel math prettyprint qw sequences ; IN: rosetta-code.metallic-ratios

lucas ( n a b -- n a' b' ) tuck reach * + ;
lucas. ( n -- )
   1 pprint bl 1 1 14 [ lucas over pprint bl ] times 3drop nl ;
approx ( a b -- d ) swap [ 0 <decimal> ] bi@ 32 D/ ;
approximate ( n -- value iter )
   -1 swap 1 1 0 1 [ 2dup = ]
   [ [ 1 + ] 5 ndip [ lucas 2dup approx ] 2dip drop ] until
   4nip decimal>ratio swap ;

qw{ 

   Platinum Golden Silver Bronze Copper Nickel Aluminum Iron
   Tin Lead

} [ 

   dup dup approximate {
       [ "Lucas sequence for %s ratio " printf ]
       [ "where b = %d:\n" printf ]
       [ "First 15 elements: " write lucas. ]
       [ "Approximated value: %.32f " printf ]
       [ "- reached after  %d  iteration(s)\n\n" printf ]
   } spread

] each-index</lang>

Output:
Lucas sequence for Platinum ratio where b = 0:
First 15 elements: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 
Approximated value: 1.00000000000000000000000000000000 - reached after  1  iteration(s)

Lucas sequence for Golden ratio where b = 1:
First 15 elements: 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 
Approximated value: 1.61803398874989484820458683436563 - reached after  78  iteration(s)

Lucas sequence for Silver ratio where b = 2:
First 15 elements: 1 1 3 7 17 41 99 239 577 1393 3363 8119 19601 47321 114243 
Approximated value: 2.41421356237309504880168872420969 - reached after  44  iteration(s)

Lucas sequence for Bronze ratio where b = 3:
First 15 elements: 1 1 4 13 43 142 469 1549 5116 16897 55807 184318 608761 2010601 6640564 
Approximated value: 3.30277563773199464655961063373524 - reached after  32  iteration(s)

Lucas sequence for Copper ratio where b = 4:
First 15 elements: 1 1 5 21 89 377 1597 6765 28657 121393 514229 2178309 9227465 39088169 165580141 
Approximated value: 4.23606797749978969640917366873127 - reached after  27  iteration(s)

Lucas sequence for Nickel ratio where b = 5:
First 15 elements: 1 1 6 31 161 836 4341 22541 117046 607771 3155901 16387276 85092281 441848681 2294335686 
Approximated value: 5.19258240356725201562535524577016 - reached after  24  iteration(s)

Lucas sequence for Aluminum ratio where b = 6:
First 15 elements: 1 1 7 43 265 1633 10063 62011 382129 2354785 14510839 89419819 551029753 3395598337 20924619775 
Approximated value: 6.16227766016837933199889354443271 - reached after  22  iteration(s)

Lucas sequence for Iron ratio where b = 7:
First 15 elements: 1 1 8 57 407 2906 20749 148149 1057792 7552693 53926643 385039194 2749201001 19629446201 140155324408 
Approximated value: 7.14005494464025913554865124576351 - reached after  20  iteration(s)

Lucas sequence for Tin ratio where b = 8:
First 15 elements: 1 1 9 73 593 4817 39129 317849 2581921 20973217 170367657 1383914473 11241683441 91317382001 741780739449 
Approximated value: 8.12310562561766054982140985597407 - reached after  19  iteration(s)

Lucas sequence for Lead ratio where b = 9:
First 15 elements: 1 1 10 91 829 7552 68797 626725 5709322 52010623 473804929 4316254984 39320099785 358197153049 3263094477226 
Approximated value: 9.10977222864644365500113714088139 - reached after  18  iteration(s)

Go

<lang go>package main

import ( 

   "fmt"
   "math/big"

)

var names = [10]string{"Platinum", "Golden", "Silver", "Bronze", "Copper", 

   "Nickel", "Aluminium", "Iron", "Tin", "Lead"}

func lucas(b int64) { 

   fmt.Printf("Lucas sequence for %s ratio, where b = %d:\n", names[b], b)
   fmt.Print("First 15 elements: ")
   var x0, x1 int64 = 1, 1
   fmt.Printf("%d, %d", x0, x1)
   for i := 1; i <= 13; i++ {
       x2 := b*x1 + x0
       fmt.Printf(", %d", x2)
       x0, x1 = x1, x2
   }
   fmt.Println()

}

func metallic(b int64, dp int) { 

   x0, x1, x2, bb := big.NewInt(1), big.NewInt(1), big.NewInt(0), big.NewInt(b)
   ratio := big.NewRat(1, 1)
   iters := 0
   prev := ratio.FloatString(dp)
   for {
       iters++
       x2.Mul(bb, x1)
       x2.Add(x2, x0)
       this := ratio.SetFrac(x2, x1).FloatString(dp)
       if prev == this {
           plural := "s"
           if iters == 1 {
               plural = " "
           }
           fmt.Printf("Value to %d dp after %2d iteration%s: %s\n\n", dp, iters, plural, this)
           return
       }
       prev = this
       x0.Set(x1)
       x1.Set(x2)
   }

}

func main() { 

   for b := int64(0); b < 10; b++ {
       lucas(b)
       metallic(b, 32)
   }
   fmt.Println("Golden ratio, where b = 1:")
   metallic(1, 256)

}</lang>

Output:
Lucas sequence for Platinum ratio, where b = 0:
First 15 elements: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
Value to 32 dp after  1 iteration : 1.00000000000000000000000000000000

Lucas sequence for Golden ratio, where b = 1:
First 15 elements: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610
Value to 32 dp after 78 iterations: 1.61803398874989484820458683436564

Lucas sequence for Silver ratio, where b = 2:
First 15 elements: 1, 1, 3, 7, 17, 41, 99, 239, 577, 1393, 3363, 8119, 19601, 47321, 114243
Value to 32 dp after 44 iterations: 2.41421356237309504880168872420970

Lucas sequence for Bronze ratio, where b = 3:
First 15 elements: 1, 1, 4, 13, 43, 142, 469, 1549, 5116, 16897, 55807, 184318, 608761, 2010601, 6640564
Value to 32 dp after 34 iterations: 3.30277563773199464655961063373525

Lucas sequence for Copper ratio, where b = 4:
First 15 elements: 1, 1, 5, 21, 89, 377, 1597, 6765, 28657, 121393, 514229, 2178309, 9227465, 39088169, 165580141
Value to 32 dp after 28 iterations: 4.23606797749978969640917366873128

Lucas sequence for Nickel ratio, where b = 5:
First 15 elements: 1, 1, 6, 31, 161, 836, 4341, 22541, 117046, 607771, 3155901, 16387276, 85092281, 441848681, 2294335686
Value to 32 dp after 25 iterations: 5.19258240356725201562535524577016

Lucas sequence for Aluminium ratio, where b = 6:
First 15 elements: 1, 1, 7, 43, 265, 1633, 10063, 62011, 382129, 2354785, 14510839, 89419819, 551029753, 3395598337, 20924619775
Value to 32 dp after 23 iterations: 6.16227766016837933199889354443272

Lucas sequence for Iron ratio, where b = 7:
First 15 elements: 1, 1, 8, 57, 407, 2906, 20749, 148149, 1057792, 7552693, 53926643, 385039194, 2749201001, 19629446201, 140155324408
Value to 32 dp after 22 iterations: 7.14005494464025913554865124576352

Lucas sequence for Tin ratio, where b = 8:
First 15 elements: 1, 1, 9, 73, 593, 4817, 39129, 317849, 2581921, 20973217, 170367657, 1383914473, 11241683441, 91317382001, 741780739449
Value to 32 dp after 20 iterations: 8.12310562561766054982140985597408

Lucas sequence for Lead ratio, where b = 9:
First 15 elements: 1, 1, 10, 91, 829, 7552, 68797, 626725, 5709322, 52010623, 473804929, 4316254984, 39320099785, 358197153049, 3263094477226
Value to 32 dp after 20 iterations: 9.10977222864644365500113714088140

Golden ratio, where b = 1:
Value to 256 dp after 615 iterations: 1.6180339887498948482045868343656381177203091798057628621354486227052604628189024497072072041893911374847540880753868917521266338622235369317931800607667263544333890865959395829056383226613199282902678806752087668925017116962070322210432162695486262963136144

Perl 6

Works with: Rakudo version 2019.07.1

Note: Arrays, Lists and Sequences are zero indexed in Perl 6.

<lang perl6>use Rat::Precise; use Lingua::EN::Numbers;

sub lucas ($b) { 1, 1, * + $b * * … * }

sub metallic ($seq, $places = 32) { 

   my $n = 0;
   my $last = 0;
   loop {
       my $approx = FatRat.new($seq[$n + 1], $seq[$n]);
       my $this = $approx.precise($places, :z);
       last if $this eq $last;
       $last = $this;
       $n++;
   }
   $last, $n

}

sub display ($value, $n) { 

   "Approximated value:", $value, "Reached after {$n} iterations: " ~
   "{ordinal-digit $n}/{ordinal-digit $n - 1} element."

}

for <Platinum Golden Silver Bronze Copper Nickel Aluminum Iron Tin Lead>.kv 

 -> \b, $name {
   my $lucas = lucas b;
   print "\nLucas sequence for $name ratio; where b = {b}:\nFirst 15 elements: ";
   say join ', ', $lucas[^15];
   say join ' ', display |metallic($lucas);

}

  1. Stretch goal

say join "\n", "\nGolden ratio to 256 decimal places:", display |metallic lucas(1), 256;</lang>

Output:
Lucas sequence for Platinum ratio; where b = 0:
First 15 elements: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
Approximated value: 1.00000000000000000000000000000000 Reached after 1 iterations: 1st/0th element.

Lucas sequence for Golden ratio; where b = 1:
First 15 elements: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610
Approximated value: 1.61803398874989484820458683436564 Reached after 78 iterations: 78th/77th element.

Lucas sequence for Silver ratio; where b = 2:
First 15 elements: 1, 1, 3, 7, 17, 41, 99, 239, 577, 1393, 3363, 8119, 19601, 47321, 114243
Approximated value: 2.41421356237309504880168872420970 Reached after 44 iterations: 44th/43rd element.

Lucas sequence for Bronze ratio; where b = 3:
First 15 elements: 1, 1, 4, 13, 43, 142, 469, 1549, 5116, 16897, 55807, 184318, 608761, 2010601, 6640564
Approximated value: 3.30277563773199464655961063373525 Reached after 34 iterations: 34th/33rd element.

Lucas sequence for Copper ratio; where b = 4:
First 15 elements: 1, 1, 5, 21, 89, 377, 1597, 6765, 28657, 121393, 514229, 2178309, 9227465, 39088169, 165580141
Approximated value: 4.23606797749978969640917366873128 Reached after 28 iterations: 28th/27th element.

Lucas sequence for Nickel ratio; where b = 5:
First 15 elements: 1, 1, 6, 31, 161, 836, 4341, 22541, 117046, 607771, 3155901, 16387276, 85092281, 441848681, 2294335686
Approximated value: 5.19258240356725201562535524577016 Reached after 25 iterations: 25th/24th element.

Lucas sequence for Aluminum ratio; where b = 6:
First 15 elements: 1, 1, 7, 43, 265, 1633, 10063, 62011, 382129, 2354785, 14510839, 89419819, 551029753, 3395598337, 20924619775
Approximated value: 6.16227766016837933199889354443272 Reached after 23 iterations: 23rd/22nd element.

Lucas sequence for Iron ratio; where b = 7:
First 15 elements: 1, 1, 8, 57, 407, 2906, 20749, 148149, 1057792, 7552693, 53926643, 385039194, 2749201001, 19629446201, 140155324408
Approximated value: 7.14005494464025913554865124576352 Reached after 22 iterations: 22nd/21st element.

Lucas sequence for Tin ratio; where b = 8:
First 15 elements: 1, 1, 9, 73, 593, 4817, 39129, 317849, 2581921, 20973217, 170367657, 1383914473, 11241683441, 91317382001, 741780739449
Approximated value: 8.12310562561766054982140985597408 Reached after 20 iterations: 20th/19th element.

Lucas sequence for Lead ratio; where b = 9:
First 15 elements: 1, 1, 10, 91, 829, 7552, 68797, 626725, 5709322, 52010623, 473804929, 4316254984, 39320099785, 358197153049, 3263094477226
Approximated value: 9.10977222864644365500113714088140 Reached after 20 iterations: 20th/19th element.

Golden ratio to 256 decimal places:
Approximated value:
1.6180339887498948482045868343656381177203091798057628621354486227052604628189024497072072041893911374847540880753868917521266338622235369317931800607667263544333890865959395829056383226613199282902678806752087668925017116962070322210432162695486262963136144
Reached after 615 iterations: 615th/614th element.

REXX

For this task,   the elements of the Lucas sequence are zero based. <lang rexx>/*REXX pgm computes the 1st N elements of the Lucas sequence for Metallic ratios 0──►9. */ parse arg n bLO bHI digs . /*obtain optional arguments from the CL*/ if n== | n=="," then n= 15 /*Not specified? Then use the default.*/ if bLO== | bLO=="," then bLO= 0 /* " " " " " " */ if bHI== | bHI=="," then bHI= 9 /* " " " " " " */ if digs== | digs=="," then digs= 32 /* " " " " " " */ numeric digits digs + length(.) /*specify number of decimal digs to use*/ metals= 'platinum golden silver bronze copper nickel aluminum iron tin lead' approx= 'the approximate value reached after ' /*literals that are used for SAYs. */ !.= /*the default name for a metallic ratio*/ 

        do k=0  to 9;  !.k= word(metals, k+1)   /*assign the (ten) metallic ratio names*/
        end   /*k*/

     do m=bLO   to bHI;  @.= 1;  $=  1  1       /*compute the sequence numbers & ratios*/
     r=.                                        /*the ratio  (so far).                 */
        do #=2  until r=old;     old= r         /*compute sequence numbers & the ratio.*/
                   #_1= #-1;       #_2= #-2     /*use variables for previous numbers.  */
        @.#= m * @.#_1     +     @.#_2          /*calculate a number i the sequence.   */
        if #<n  then $= $  @.#                  /*build a sequence list of  N  numbers.*/
        r= @.#  /  @.#_1                        /*calculate ratio of the last 2 numbers*/
        end   /*#*/

     if words($)<n  then $= subword($ copies('1 ', n), 1, n) /*extend list if too short*/
     L= max(108, length($) )                                 /*ensure width of title.  */
     say center(' Lucas sequence for the'  !.m  "ratio,  where  B  is " m' ',  L,  "═")
     if n>0  then do;   say 'the first '    n    " elements are:";       say $
                  end                           /*if  N  is positive, then show N nums.*/
     say approx #-1  ' iterations with '  digs  " decimal digits past the decimal point:"
     say format(r,,digs);                 say   /*display the ration plus a blank line.*/
     end      /*m*/                             /*stick a fork in it,  we're all done. */</lang>
output   when using the default inputs:
═════════════════════════ Lucas sequence for the platinum ratio,  where  B  is  0 ══════════════════════════
the first  15  elements are:
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
the approximate value reached after  2  iterations with  32  decimal digits past the decimal point:
1.00000000000000000000000000000000

══════════════════════════ Lucas sequence for the golden ratio,  where  B  is  1 ═══════════════════════════
the first  15  elements are:
1 1 2 3 5 8 13 21 34 55 89 144 233 377 610
the approximate value reached after  78  iterations with  32  decimal digits past the decimal point:
1.61803398874989484820458683436564

══════════════════════════ Lucas sequence for the silver ratio,  where  B  is  2 ═══════════════════════════
the first  15  elements are:
1 1 3 7 17 41 99 239 577 1393 3363 8119 19601 47321 114243
the approximate value reached after  44  iterations with  32  decimal digits past the decimal point:
2.41421356237309504880168872420970

══════════════════════════ Lucas sequence for the bronze ratio,  where  B  is  3 ═══════════════════════════
the first  15  elements are:
1 1 4 13 43 142 469 1549 5116 16897 55807 184318 608761 2010601 6640564
the approximate value reached after  34  iterations with  32  decimal digits past the decimal point:
3.30277563773199464655961063373525

══════════════════════════ Lucas sequence for the copper ratio,  where  B  is  4 ═══════════════════════════
the first  15  elements are:
1 1 5 21 89 377 1597 6765 28657 121393 514229 2178309 9227465 39088169 165580141
the approximate value reached after  28  iterations with  32  decimal digits past the decimal point:
4.23606797749978969640917366873128

══════════════════════════ Lucas sequence for the nickel ratio,  where  B  is  5 ═══════════════════════════
the first  15  elements are:
1 1 6 31 161 836 4341 22541 117046 607771 3155901 16387276 85092281 441848681 2294335686
the approximate value reached after  25  iterations with  32  decimal digits past the decimal point:
5.19258240356725201562535524577016

═════════════════════════ Lucas sequence for the aluminum ratio,  where  B  is  6 ══════════════════════════
the first  15  elements are:
1 1 7 43 265 1633 10063 62011 382129 2354785 14510839 89419819 551029753 3395598337 20924619775
the approximate value reached after  23  iterations with  32  decimal digits past the decimal point:
6.16227766016837933199889354443272

═══════════════════════════ Lucas sequence for the iron ratio,  where  B  is  7 ════════════════════════════
the first  15  elements are:
1 1 8 57 407 2906 20749 148149 1057792 7552693 53926643 385039194 2749201001 19629446201 140155324408
the approximate value reached after  22  iterations with  32  decimal digits past the decimal point:
7.14005494464025913554865124576352

════════════════════════════ Lucas sequence for the tin ratio,  where  B  is  8 ════════════════════════════
the first  15  elements are:
1 1 9 73 593 4817 39129 317849 2581921 20973217 170367657 1383914473 11241683441 91317382001 741780739449
the approximate value reached after  20  iterations with  32  decimal digits past the decimal point:
8.12310562561766054982140985597408

═══════════════════════════ Lucas sequence for the lead ratio,  where  B  is  9 ════════════════════════════
the first  15  elements are:
1 1 10 91 829 7552 68797 626725 5709322 52010623 473804929 4316254984 39320099785 358197153049 3263094477226
the approximate value reached after  20  iterations with  32  decimal digits past the decimal point:
9.10977222864644365500113714088140

{{out|output|text=  when using the inputs of:     0   1   1   256 

══════════════════════════ Lucas sequence for the golden ratio,  where  B  is  1 ═══════════════════════════
the approximate value reached after  615  iterations with  256  decimal digits past the decimal point:
1.6180339887498948482045868343656381177203091798057628621354486227052604628189024497072072041893911374847540880753868917521266338622235369317931800607667263544333890865959395829056383226613199282902678806752087668925017116962070322210432162695486262963136144

zkl

Library: GMP

GNU Multiple Precision Arithmetic Library

<lang zkl>var [const] BI=Import("zklBigNum"); // libGMP fcn lucasSeq(b){ 

  Walker.zero().tweak('wrap(xs){
     xm2,xm1 := xs;	// x[n-2], x[n-1]
     xn:=xm1*b + xm2;
     xs.append(xn).del(0);
     xn
  }.fp(L(BI(1),BI(1)))).push(1,1)  // xn can get big so use BigInts

} fcn metallicRatio(lucasSeq){ 

  const bige="1e33",E=10;  // x[n-1]*bige*b / x[n-2] to get our digits from Ints
  a,b := lucasSeq.next(), lucasSeq.next();
  do(1000){	// limit iterations
     c:=lucasSeq.next();
     m1,m2 := BI(bige).mul(b).div(a), BI(bige).mul(c).div(b);
     a,b = b,c;
     if((m1 - m2).abs()<E) return(m1.div(E),lucasSeq.idx); // idx ignores push(), ie first 2 terms
  }

}</lang> <lang zkl>metals:="Platinum Golden Silver Bronze Copper Nickel Aluminum Iron Tin Lead"; foreach metal in (metals.split(" ")){ n:=__metalWalker.idx; 

  println("\nLucas sequence for %s ratio; where b = %d:".fmt(metal,n));
  println("First 15 elements: ",lucasSeq(n).walk(15).concat(" "));
  mr,i := metallicRatio(lucasSeq(n));
  mr    = mr.toString();
  println("Approximated value: %s.%s - Reached after ~%d iterations."
    .fmt(mr[0],mr.del(0),i));

}</lang>

Output:
Lucas sequence for Platinum ratio; where b = 0:
First 15 elements: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
Approximated value: 1.00000000000000000000000000000000 - Reached after ~0 iterations.

Lucas sequence for Golden ratio; where b = 1:
First 15 elements: 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610
Approximated value: 1.61803398874989484820458683436564 - Reached after ~77 iterations.

Lucas sequence for Silver ratio; where b = 2:
First 15 elements: 1 1 3 7 17 41 99 239 577 1393 3363 8119 19601 47321 114243
Approximated value: 2.41421356237309504880168872420970 - Reached after ~43 iterations.

Lucas sequence for Bronze ratio; where b = 3:
First 15 elements: 1 1 4 13 43 142 469 1549 5116 16897 55807 184318 608761 2010601 6640564
Approximated value: 3.30277563773199464655961063373524 - Reached after ~32 iterations.

Lucas sequence for Copper ratio; where b = 4:
First 15 elements: 1 1 5 21 89 377 1597 6765 28657 121393 514229 2178309 9227465 39088169 165580141
Approximated value: 4.23606797749978969640917366873127 - Reached after ~27 iterations.

Lucas sequence for Nickel ratio; where b = 5:
First 15 elements: 1 1 6 31 161 836 4341 22541 117046 607771 3155901 16387276 85092281 441848681 2294335686
Approximated value: 5.19258240356725201562535524577016 - Reached after ~24 iterations.

Lucas sequence for Aluminum ratio; where b = 6:
First 15 elements: 1 1 7 43 265 1633 10063 62011 382129 2354785 14510839 89419819 551029753 3395598337 20924619775
Approximated value: 6.16227766016837933199889354443271 - Reached after ~22 iterations.

Lucas sequence for Iron ratio; where b = 7:
First 15 elements: 1 1 8 57 407 2906 20749 148149 1057792 7552693 53926643 385039194 2749201001 19629446201 140155324408
Approximated value: 7.14005494464025913554865124576351 - Reached after ~20 iterations.

Lucas sequence for Tin ratio; where b = 8:
First 15 elements: 1 1 9 73 593 4817 39129 317849 2581921 20973217 170367657 1383914473 11241683441 91317382001 741780739449
Approximated value: 8.12310562561766054982140985597407 - Reached after ~19 iterations.

Lucas sequence for Lead ratio; where b = 9:
First 15 elements: 1 1 10 91 829 7552 68797 626725 5709322 52010623 473804929 4316254984 39320099785 358197153049 3263094477226
Approximated value: 9.10977222864644365500113714088139 - Reached after ~18 iterations.
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