Visualising Recamán's sequence using TikZ

I am trying to visualise Recamán's sequence using semicircles. This is an example of what I'm trying to replicate:

By manually defining each semicircle, I've got this so far:

documentclass[tikz,margin=0.5cm]standalone
begindocument
begintikzpicture
draw [line width=3mm] (0,0) arc (-180:0:1/2) arc (180:0:2/2) arc (-180:0:3/2) arc (-180:0:-4/2) arc (-180:0:5/2) arc (180:0:6/2) arc (-180:0:7/2) arc (-180:0:-8/2) arc (-180:0:9/2) arc (-180:0:-10/2);
endtikzpicture
enddocument


Is there a way to code this so it will automatically generate N iterations of the sequence?

In basic terms, the size of the steps increases sequentially, but you subtract if you can (i.e. if you arrive at a number that hasn't been used before), otherwise you add.

For reference, this sequence is A005132 on 'The On-Line Encyclopedia of Integer Sequences' - https://oeis.org/A005132

2 Answers
2

As I am quite ignorant in TikZ I can't add the colours etc...

first we generate the first 1000 sequence members,

second, we use some kind of expandable loop in the tikz picture. For this I loaded xinttools, but there are other choices.

xinttools

third I did three separate pictures but one can probably use in the same picture successive loops with new colors perhaps, or vary the color with index #1.

I did not abstract into a macro with the number of steps as argument, and I did not do the work to created animated gif, but this could give a start.

Also, I spend a bit time in TikZ manual looking for half-circles, but I got confused, so I stick with the arc construct of OP, I don't know if efficient or not for TikZ matters.

arc

documentclass[tikz,margin=0.5cm]standalone

% FIRST WE GENERATE a(n) for n=0, ..., 1000
% https://oeis.org/A005132
makeatletter
@namedefrecaman00@namedefnamacer00
@namedefrecaman11@namedefnamacer11
@namedefrecaman23@namedefnamacer32
@namedefrecaman36@namedefnamacer63
% <namacerN> will give the *last* index n with a(n) = N

count@ 3
loop
advancecount@ @ne
edefzzzthenumexpr
@nameuserecamanthenumexprcount@-@ne-count@%
@namedefrecagoleftthecount@0%
ifnumzzz>z@
ifcsname namacerzzzendcsname
edefzzzthenumexprzzz+count@+count@%
else
@namedefrecagoleftthecount@1%
fi
else
edefzzzthenumexprzzz+count@+count@%
fi
expandafteredefcsname recamanthecount@endcsnamezzz%
expandafteredefcsname namacerzzzendcsnamethecount@%
typeouta(thecount@) = @nameuserecamanthecount@%
ifnumcount@<1000
repeat

usepackagexinttools

makeatletter
% I added the "recagoleft" in a second stage originally I was doing
% ifnum test to check if increase or decrease for index 2*#1 and
% 2*#1+1
% (perhaps I should have kept more cumbersome ifnum rather than
% creating these extra macros?)
defmymacro#1%
arc
if@nameuserecagoleftthenumexpr2*#11%
expandafter@firstoftwoelseexpandafter@secondoftwofi
(-180:0:-thenumexpr2*#1relax)(180:0:thenumexpr2*#1relax)%
arc
if@nameuserecagoleftthenumexpr2*#1+11%
expandafter@firstoftwoelseexpandafter@secondoftwofi
(180:0:-thenumexpr2*#1+1relax)(-180:0:thenumexpr2*#1+1relax)%

makeatother

begindocument
begintikzpicture
draw [line width=3mm]
(0,0) arc
(-180:0:1)
xintApplyUnbracedmymacroxintSeq110;
endtikzpicture
begintikzpicture
draw [line width=3mm]
(0,0) arc
(-180:0:1)
xintApplyUnbracedmymacroxintSeq130;
endtikzpicture
begintikzpicture
draw [line width=3mm]
(0,0) arc
(-180:0:1)
xintApplyUnbracedmymacroxintSeq150;
endtikzpicture

enddocument


For generating the picture I had the problem that my usual gs invocation (with transparent background) proved very slow and produced very big png's... (13M for the third one). So I did it again with smaller resolution and here is the first one (for n=2*10=20, circa).

For the one up to n=2*50+1=101 I upload a screen capture

I realized only later that 50(i.e. n=101) was maximal with 1cm units before triggering "dimension too large error" from TikZ. But this works, where the unit is about 1.5pt (2pt would be too big).

50

1cm

1.5pt

2pt

begintikzpicture[x=100000sp,y=100000sp]
draw [line width=3mm]
(0,0) arc
(-180:0:1)
xintApplyUnbracedmymacroxintSeq1499;% and not 500 as a(1001) not pre-computed
endtikzpicture


I obtain this, which indicates the values of sequence up to n=999: (sorry I initially uploaded a possibly wrong picture because I used 500, hence n=1001 but my pre-computations in preamble go only up to 1000; no error was triggered because only an if test is done about some control sequence, not a computation or an ifnum)

n=999

if

ifnum

I am almost an expert in TikZ now: a draw statement can have only one colour. So I modified my approach to use some loop to accumulate multiple draw with a colour from an xcolor color series.

draw

draw

xcolor

I wanted to create an animated gif, for this reason I do a super-loop which increases the number of steps each time. My usual convert invocation failed, using seemingly the size of the first pdf page. Then I modified the code to use the same picture size for all frames. This is why in the code below I added new macros holding max(a(n), i=0..n).

convert

max(a(n), i=0..n)

But convert again failed so I post only here a snapshot of the last frame.

convert

But if you compile to pdf, your PDF viewer possibly with re-create the animation for you by holding down the space key (it works for me; I configured view to a single page i.e. not "continous pages").

documentclass[tikz]standalone
%usepackagexcolor
% FIRST WE GENERATE a(n) for n=0, ..., 1000
% https://oeis.org/A005132
makeatletter
@namedefrecaman00@namedefnamacer00
@namedefrecaman11@namedefnamacer11@namedefrecagoleft10
@namedefrecaman23@namedefnamacer32@namedefrecagoleft20
@namedefrecaman36@namedefnamacer63@namedefrecagoleft30
% <namacerN> will give the *last* index n with a(n) = N
@namedefrecaMax00
@namedefrecaMax11
@namedefrecaMax23
@namedefrecaMax36
count@ 3
loop
advancecount@ @ne
edefzzzthenumexpr
@nameuserecamanthenumexprcount@-@ne-count@%
@namedefrecagoleftthecount@0%
expandafterletcsname recaMaxthecount@expandafterendcsname
csname recaMaxthenumexprcount@-@neendcsname
ifnumzzz>z@
ifcsname namacerzzzendcsname
edefzzzthenumexprzzz+count@+count@%
ifnumzzz>@nameuserecaMaxthecount@
expandafterletcsname recaMaxthecount@endcsnamezzz
fi
else
@namedefrecagoleftthecount@1%
fi
else
edefzzzthenumexprzzz+count@+count@%
ifnumzzz>@nameuserecaMaxthecount@
expandafterletcsname recaMaxthecount@endcsnamezzz
fi
fi
expandafterletcsname recamanthecount@endcsnamezzz
expandafteredefcsname namacerzzzendcsnamethecount@%
typeouta(thecount@) = @nameuserecamanthecount@
(max so far=@nameuserecaMaxthecount@)%
ifnumcount@<1000
repeat

usepackagexinttools

makeatletter
defmymacro#1%
draw
[color=foo!!+]
% radius being n, width of circle 2n, end-point is at 2*a(n)
(2*@nameuserecamanthenumexpr#1-1,0)
arc
if@nameuserecagoleft#11%
expandafter@firstoftwoelseexpandafter@secondoftwofi
(unlessifodd#1 -fi180:0:-#1)(ifodd#1 -fi180:0:#1);%

makeatother

makeatletter
defdrawframe#1noexpand
draw
(0,0)--(0,-#1)--(thenumexpr2*@nameuserecaMax#1,-#1)
--(thenumexpr2*@nameuserecaMax#1,#1)--(0,#1)--cycle;%
makeatother

definecolorseriesfoorgblastbluered

begindocument

xintFor* #1 in xintSeq1100do%
begintikzpicture[x=5mm, y=5mm]
edefzzzdrawframe100zzz% get all pictures to be of same size
resetcolorseries[#1]foo%
xintApplyUnbracedmymacroxintSeq1#1
endtikzpicture


enddocument


Here is thus page 100 of the produced PDF:

I finally manage to get an animated gif:

Recipe:

[x=1mm, y=1mm]

line width=.5mm

convert -density 72 recaman-colors.pdf _tmp%02d.png

convert -verbose -dispose previous -loop 0 -density 100 -delay 15 _tmp00..9.png _tmp10..63.png -delay 300 _tmp64.png recaman.gif

Output not as smooth as one could hope, but 360383 (was for 50 frames) 745256 bytes.

I'm sorry, really thought these should be spirals. Should have checked. Sorry! With insert path you can, of course, insert whatever you like, also a series of arcs. I have not much time now, unfortunately ...

documentclass[tikz,border=3.14mm]standalone
tikzsetRecaman/.style n args=3insert path=
foreach X in #1,...,#2
arc (-180:0:#3*(2*X-1)/2) arc (0:180:#3*2*X/2)

begindocument
begintikzpicture
expandafterdraw[rounded corners] (0,1) to[out=0,in=90] (3,0)
[Recaman=260.3] arc(-180:0:2) to[out=90,in=90] ++(3,1)
[Recaman=270.2];
endtikzpicture
enddocument


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