Hofstadter Figure-Figure sequences: Difference between revisions

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These two sequences of positive integers are defined as:

:<math>\begin{align}

:::: <big><big><math> \begin{align}

R(1)&=1\ ;\ ~~S(1)=2 \\~~

R(1)&=1\ \\

R(n)&=~~R(n-1)+S(n-1),~~ \~~quad~~ ~~n>1.~~

S(1)&=2\ \\

R(n)&=R(n-1)+S(n-1), \quad n>1.

\end{align}</math>

\end{align} </math></big></big>

⚫

:The sequence <math>S(n)</math> is further defined as the sequence of positive integers ''not'' present in <math>R(n)</math>.

⚫

~~Sequence~~ R ~~starts:~~ 1, 3, 7, 12, 18, ...~~ ~~

⚫

~~Sequence~~ S ~~starts:~~ 2, 4, 5, 6, 8, ...

⚫

The sequence   <big><big><math> S(n) </math></big></big>   is further defined as the sequence of positive integers   ''not''   present in   <big><big><math> R(n).</math></big></big>

Sequence   <big><big><math> R </math></big></big>   starts:

⚫

1, 3, 7, 12, 18, ...

Sequence   <big><big><math> S </math></big></big>   starts:

⚫

2, 4, 5, 6, 8, ...

;Task:

# Create two functions named ffr and ffs that when given n return R(n) or S(n) respectively.<br~~><small~~>(Note that R(1) = 1 and S(1) = 2 to avoid off-by-one errors).~~~~

# Create two functions named   <big> '''ffr''' </big>   and   <big> '''ffs''' </big>   that when given   <big><big><math> n </math></big></big>   return   <big><big><math> R(n) </math></big></big>   or   <big><big><math> S(n) </math></big></big>   respectively. (Note that   R(1) = 1   and   S(1) = 2   to avoid off-by-one errors).

# No maximum value for n should be assumed.

# No maximum value for   <big><big><math> n </math></big></big>   should be assumed.

# Calculate and show that the first ten values of R are: 1, 3, 7, 12, 18, 26, 35, 45, 56, and 69

# Calculate and show that the first ten values of   <big><big><math> R </math></big></big>   are: 1, 3, 7, 12, 18, 26, 35, 45, 56, and 69

# Calculate and show that the first 40 values of ffr plus the first 960 values of ffs include all the integers from 1 to 1000 exactly once.

# Calculate and show that the first 40 values of   <big> '''ffr''' </big>   plus the first 960 values of   <big> '''ffs''' </big>   include all the integers from '''1''' to '''1000''' exactly once.

;References:

* Sloane's [http://oeis.org/A005228 A005228] and [http://oeis.org/A030124 A030124].

* [http://mathworld.wolfram.com/HofstadterFigure-FigureSequence.html Wolfram ~~Mathworld~~]

* [http://mathworld.wolfram.com/HofstadterFigure-FigureSequence.html Wolfram MathWorld]

* Wikipedia: [[wp:Hofstadter_sequence#Hofstadter_Figure-Figure_sequences|Hofstadter Figure-Figure sequences]].

=={{header|Ada}}==

Specifying a package providing the functions FFR and FFS: