Hofstadter Figure-Figure sequences: Difference between revisions

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These two sequences of positive integers are defined as:
These two sequences of positive integers are defined as:
:::: <big><big><math> \begin{align}
:::: <big><math>\begin{align}
R(1)&=1\ \\
R(1)&=1\ ;\ S(1)=2 \\
S(1)&=2\ \\
R(n)&=R(n-1)+S(n-1), \quad n>1.
⚫
\end{align}</math></big>
R(n)&=R(n-1)+S(n-1), \quad n>1.
⚫
\end{align} </math></big></big>


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


Sequence &nbsp; <big><big><math> R </math></big></big> &nbsp; starts:
Sequence <big><math>R</math></big> starts:
1, 3, 7, 12, 18, ...
1, 3, 7, 12, 18, ...
Sequence &nbsp; <big><big><math> S </math></big></big> &nbsp; starts:
Sequence <big><math>S</math></big> starts:
2, 4, 5, 6, 8, ...
2, 4, 5, 6, 8, ...




;Task:
;Task:
# Create two functions named &nbsp; <big> '''ffr''' </big> &nbsp; and &nbsp; <big> '''ffs''' </big> &nbsp; that when given &nbsp; <big><big><math> n </math></big></big> &nbsp; return &nbsp; <big><big><math> R(n) </math></big></big> &nbsp; or &nbsp; <big><big><math> S(n) </math></big></big> &nbsp; respectively. <br>(Note that &nbsp; R(1) = 1 &nbsp; and &nbsp; S(1) = 2 &nbsp; to avoid off-by-one errors).
# Create two functions named '''ffr''' and '''ffs''' that when given '''n''' return '''R(n)''' or '''S(n)''' respectively.<br>(Note that R(1) = 1 and S(1) = 2 to avoid off-by-one errors).
# No maximum value for &nbsp; <big><big><math> n </math></big></big> &nbsp; should be assumed.
# No maximum value for '''n''' should be assumed.
# Calculate and show that the first ten values of &nbsp; <big><big><math> R </math></big></big> &nbsp; are: <br>1, 3, 7, 12, 18, 26, 35, 45, 56, and 69
# Calculate and show that the first ten values of '''R''' are:<br> 1, 3, 7, 12, 18, 26, 35, 45, 56, and 69
# Calculate and show that the first 40 values of &nbsp; <big> '''ffr''' </big> &nbsp; plus the first 960 values of &nbsp; <big> '''ffs''' </big> &nbsp; include all the integers from '''1''' to '''1000''' exactly once.
# 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.




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