Otdfbt

How can I make number pattern like this

But, all spaces are replaced with plus signs ?

Just for fun: without packages.

documentclassarticle
newcounterpft
newcounterpfft
begindocument
setcounterpft0loop%
stepcounterpft%
setcounterpfft0noindent$loop%
stepcounterpfftnumbervaluepfft%
ifnumnumbervaluepfft<numbervaluepft+repeat$par%
ifnumnumbervaluepft<11repeat%
enddocument

Or with pgffor, which contains the foreach command that is used in JouleV's nice answer, and which is loaded by TikZ.

documentclassarticle
usepackagepgffor
begindocument
foreach X in 1,...,11%
noindent$foreach Y in 1,...,X%
ifnumY>1%
+%
fi%
Y
$par
enddocument

Since everyone has a bonus...

documentclassarticle
newcounterpft
newcounterpfft
newcountersum
begindocument
beginflushright
setcounterpft0loop%
stepcounterpft%
setcounterpfft0setcountersum0noindent$loop%
stepcounterpfftnumbervaluepfftaddtocountersumvaluepfft%
ifnumnumbervaluepfft<numbervaluepft+repeat=numbervaluesum%
ifnumvaluesum<10relaxphantom1fi$par%
ifnumnumbervaluepft<11repeat%
endflushright
enddocument

Let's employ TikZ and some simple algorithms for such things.

documentclass[tikz]standalone
begindocument
begintikzpicture[x=5ex,y=5ex]
defmaxnum11 % Change the size here
foreach i in 1,...,maxnum 
 % Draw the numbers
 pgfmathsetmacroauxnummaxnum-i+1
 foreach j in 1,...,auxnum 
 node (i-j) at (i,j) i;
 % Draw the plus signs
 ifnumi=maxnumrelaxelse
 pgfmathsetmacroauxnummaxnum-i
 foreach j in 1,...,auxnum 
 node[anchor=base] at ([shift=(.5,0)]i-j.base) +;
 fi

endtikzpicture
enddocument

A little bonus:

documentclass[tikz]standalone
begindocument
begintikzpicture[x=5ex,y=5ex]
defmaxnum11
foreach i in 1,...,maxnum 
 % Draw the numbers
 pgfmathsetmacroauxnummaxnum-i+1
 foreach j in 1,...,auxnum 
 node (i-j) at (i,j) i;
 % Draw the plus signs
 ifnumi=maxnumrelaxelse
 pgfmathsetmacroauxnummaxnum-i
 foreach j in 1,...,auxnum 
 node[anchor=base] at ([shift=(.5,0)]i-j.base) +;
 fi
 % The rest :)
 pgfmathsetmacrosumoutputint((maxnum-i+1)*(maxnum-i+2)/2)
 node (output-i) at (maxnum+1,i) sumoutput;
 node[anchor=base] at ([shift=(-.5,0)]output-i.base) =;

endtikzpicture
enddocument

Not sure what do you really want to get by "to the right", but maybe this?

documentclass[tikz]standalone
begindocument
begintikzpicture[x=5ex,y=5ex]
defmaxnum11
foreach i in 1,...,maxnum 
 % numbers - simpler than the above code
 foreach j [count=k] in i,...,1
 node (i-j) at (i,j) k;
 % plus signs - a bit simpler
 ifnumi=maxnumrelaxelse
 foreach j in i,...,1 
 node[anchor=base] at ([shift=(.5,0)]i-j.base) +;
 fi
 % I keep the "bonus" part - it needs no improvements
 pgfmathsetmacrosumoutputint((maxnum-i+1)*(maxnum-i+2)/2)
 node (output-i) at (maxnum+1,i) sumoutput;
 node[anchor=base] at ([shift=(-.5,0)]output-i.base) =;

endtikzpicture
enddocument

Comparison with other answers

Other answers are very good, but it seems that this answer is totally different from all other answers (which are somehow a little similar to each other). This is the pros and cons of this answer compared to the others. Read it to consider what to use.

Pros

The numbers and signs are aligned in a perfect square. So you can have something like this

documentclass[tikz]standalone
usetikzlibrarybackgrounds,fit,calc
begindocument
begintikzpicture[x=5ex,y=5ex]
defmaxnum11 % Change the size here
foreach i in 1,...,maxnum 
 % numbers - simpler than the above code
 foreach j [count=k] in i,...,1
 node[minimum size=.55cm] (i-j) at (i,j) k;
 % plus signs - a bit simpler
 ifnumi=maxnumrelaxelse
 foreach j in i,...,1 
 node[anchor=base] at ([shift=(.5,0)]i-j.base) +;
 fi
 % I keep the "bonus" part - it needs no improvements
 pgfmathsetmacrosumoutputint((maxnum-i+1)*(maxnum-i+2)/2)
 node (output-i) at (maxnum+1,i) sumoutput;
 node[anchor=base] at ([shift=(-.5,0)]output-i.base) =;

beginscope[on background layer]
path[fill=red!10,draw=red,thick] 
 ([shift=( .1,-.1)]1-1.north west) -- 
 ([shift=( .1,-.1)]11-11.north west) 
 arc (135:-45:.55/sqrt(2)) --
 ([shift=(-.1, .1)]1-1.south east) 
 arc (315:135:.55/sqrt(2)); % Well, possibly this is not the best
 % way to draw this, but this is only
 % for illustration purpose.
draw[red,thick,<-] 
 ($([shift=( .1,-.1)]1-1.north west)!.45!([shift=( .1,-.1)]11-11.north west)$)
 -- + (-2,1) node[pos=1.3] Looks! All are number 1!;
endscope
endtikzpicture
enddocument

In other answers, the numbers, say 1, are not perfectly aligned if maxnum is greater than 9, so it is harder to draw such a path.

Cons

It is very clear: the spacing between the numbers and the plus signs are inconsistent. But when I follow this approach (insert the plus signs separately), I have to accept this kind of spacing.

Btw, the numbers after the equal signs are not right-aligned. This is fixable, but a fix will make the code much more complicated (if I want to automatically determine the width of the node of the "sum"). I don't think this is a major problem though.

Here's a LuaLaTeX-based solution. The number of rows can be set as the argument of the LaTeX macro makearray; this macro calls the Lua function make_array, which does most of the work.

To make the triangular array right-aligned instead of left-aligned, change beginarray@l@ to beginarray@r@. If you wish to have a bit more space around the + symbols, change tex.sprint ( "+" ) to tex.sprint ( "+" ).

% !TEX TS-program = lualatex
documentclassarticle
usepackageluacode % for 'luacode' environment

%% Lua-side code: Define the Lua function 'make_array'
beginluacode

function make_array ( n )
 for i=1,n do
 for j=1,i do
 tex.sprint ( j )
 if j<i then 
 tex.sprint ( "+" )
 else
 tex.sprint ( "\\" )
 end
 end
 end
end

endluacode

%% LaTeX-side code: Define the macro 'makearray'
newcommandmakearray[1]%
 $beginarray@l@
 directluamake_array(#1)
 endarray$


begindocument
makearray11
enddocument

Addendum: Here's a solution which typesets each row's sum along the right-hand edge of the (now rectangular) array. Compared with the solution shown above, it also allows for a bit more space around the + symbols. Observe the use of toprule and bottomrule: These instruction are optional and may be left off.

% !TEX TS-program = lualatex
documentclassarticle
usepackageluacode % for 'luaexec' macro
usepackagebooktabs % for 'toprule' and 'bottomrule' macros

%% Lua-side code: Define the function 'make_array_with_rowsum'
luaexec%

function make_array_with_rowsum ( n )
 for i=1,n do
 for j=1,i do
 tex.sprint ( j )
 if j<i then 
 tex.sprint ( "+" )
 else
 tex.sprint ( "&=&" .. i*(i+1)//2 .. "\\" )
 end
 end
 end
end

%% LaTeX-side code: Define the macro 'makearray'
newcommandmakearray[1]%
 parnoindent
 begingroup
 centering
 setlengtharraycolsep0pt
 $beginarraylcr
 toprule
 luadirectmake_array_with_rowsum(#1)
 bottomrule
 endarray$par
 endgroup

begindocument
makearray16
enddocument

documentclassarticle
usepackagemultido
begindocument
noindent$1$
multidoiA=2+110\$%
 multidoiB=1+1numexpriA-1iB+iA$

bigskipnoindent$1=1$
multidoiA=2+110\$xdefSum0%
 multidoiB=1+1numexpriA-1%
 iB+xdefSumthenumexprSum+iBiA=thenumexprSum+iA$

enddocument

The mandatory double expl3 loop:

documentclassarticle
usepackagexparse,array

ExplSyntaxOn
NewDocumentCommandtriangletabsm
 % #1 = * for totals, #2 = end point
 IfBooleanTF #1 
 
 begintabular@ l @ c @ r @
 int_step_inline:nn #2 panda_row_total:n ##1 \ 
 endtabular
 
 
 begintabular@ l @
 int_step_inline:nn #2 panda_row:n ##1 \ 
 endtabular
 
 
cs_new_protected:Nn panda_row:n
 
 $1 int_step_inline:nnn 2 #1 + ##1 $
 
cs_new_protected:Nn panda_row_total:n
 
 panda_row:n #1 & $=$ & $int_eval:n (#1)*(#1+1)/2 $
 
ExplSyntaxOff

begindocument

[
triangletab1 qquad triangletab3 qquad triangletab11
]

[
triangletab*1 qquad triangletab*3 qquad triangletab*11
]

enddocument