Catalan numbers: Difference between revisions
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=={{header|Phix}}== |
=={{header|Phix}}== |
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See also [[Catalan_numbers/Pascal%27s_triangle#Phix]] which may be faster. |
See also [[Catalan_numbers/Pascal%27s_triangle#Phix]] which may be faster. |
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<!--<lang Phix>--> |
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<lang Phix>-- returns inf/-nan for n>85, and needs the rounding for n>=14, accurate to n=29 |
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<span style="color: #000080;font-style:italic;">-- returns inf/-nan for n>85, and needs the rounding for n>=14, accurate to n=29</span> |
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function catalan1(integer n) |
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<span style="color: #008080;">function</span> <span style="color: #000000;">catalan1</span><span style="color: #0000FF;">(</span><span style="color: #004080;">integer</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">)</span> |
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return floor(factorial(2*n)/(factorial(n+1)*factorial(n))+0.5) |
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<span style="color: #008080;">return</span> <span style="color: #7060A8;">floor</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">factorial</span><span style="color: #0000FF;">(</span><span style="color: #000000;">2</span><span style="color: #0000FF;">*</span><span style="color: #000000;">n</span><span style="color: #0000FF;">)/(</span><span style="color: #7060A8;">factorial</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">)*</span><span style="color: #7060A8;">factorial</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">))+</span><span style="color: #000000;">0.5</span><span style="color: #0000FF;">)</span> |
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end function |
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<span style="color: #008080;">end</span> <span style="color: #008080;">function</span> |
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-- returns inf for n>519, accurate to n=30: |
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<span style="color: #000080;font-style:italic;">-- returns inf for n>519, accurate to n=30:</span> |
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function catalan2(integer n) -- NB: very slow! |
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<span style="color: #008080;">function</span> <span style="color: #000000;">catalan2</span><span style="color: #0000FF;">(</span><span style="color: #004080;">integer</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">)</span> <span style="color: #000080;font-style:italic;">-- NB: very slow!</span> |
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atom res = not n |
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<span style="color: #004080;">atom</span> <span style="color: #000000;">res</span> <span style="color: #0000FF;">=</span> <span style="color: #008080;">not</span> <span style="color: #000000;">n</span> |
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n -= 1 |
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<span style="color: #000000;">n</span> <span style="color: #0000FF;">-=</span> <span style="color: #000000;">1</span> |
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for i=0 to n do |
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<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">0</span> <span style="color: #008080;">to</span> <span style="color: #000000;">n</span> <span style="color: #008080;">do</span> |
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res += catalan2(i)*catalan2(n-i) |
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<span style="color: #000000;">res</span> <span style="color: #0000FF;">+=</span> <span style="color: #000000;">catalan2</span><span style="color: #0000FF;">(</span><span style="color: #000000;">i</span><span style="color: #0000FF;">)*</span><span style="color: #000000;">catalan2</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">-</span><span style="color: #000000;">i</span><span style="color: #0000FF;">)</span> |
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end for |
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<span style="color: #008080;">end</span> <span style="color: #008080;">for</span> |
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return res |
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<span style="color: #008080;">return</span> <span style="color: #000000;">res</span> |
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end function |
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<span style="color: #008080;">end</span> <span style="color: #008080;">function</span> |
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-- returns inf for n>514, accurate to n=30: |
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<span style="color: #000080;font-style:italic;">-- returns inf for n>514, accurate to n=30:</span> |
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function catalan3(integer n) |
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<span style="color: #008080;">function</span> <span style="color: #000000;">catalan3</span><span style="color: #0000FF;">(</span><span style="color: #004080;">integer</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">)</span> |
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if n=0 then return 1 end if |
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<span style="color: #008080;">if</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">=</span><span style="color: #000000;">0</span> <span style="color: #008080;">then</span> <span style="color: #008080;">return</span> <span style="color: #000000;">1</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span> |
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return 2*(2*n-1)/(1+n)*catalan3(n-1) |
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<span style="color: #008080;">return</span> <span style="color: #000000;">2</span><span style="color: #0000FF;">*(</span><span style="color: #000000;">2</span><span style="color: #0000FF;">*</span><span style="color: #000000;">n</span><span style="color: #0000FF;">-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">)/(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">+</span><span style="color: #000000;">n</span><span style="color: #0000FF;">)*</span><span style="color: #000000;">catalan3</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">)</span> |
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end function |
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<span style="color: #008080;">end</span> <span style="color: #008080;">function</span> |
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sequence res = repeat(repeat(0,4),16), |
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<span style="color: #004080;">sequence</span> <span style="color: #000000;">res</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">repeat</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">repeat</span><span style="color: #0000FF;">(</span><span style="color: #000000;">0</span><span style="color: #0000FF;">,</span><span style="color: #000000;">4</span><span style="color: #0000FF;">),</span><span style="color: #000000;">16</span><span style="color: #0000FF;">),</span> |
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times = repeat(0,3) |
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<span style="color: #000000;">times</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">repeat</span><span style="color: #0000FF;">(</span><span style="color: #000000;">0</span><span style="color: #0000FF;">,</span><span style="color: #000000;">3</span><span style="color: #0000FF;">)</span> |
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for t=1 to 4 do |
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<span style="color: #008080;">for</span> <span style="color: #000000;">t</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">4</span> <span style="color: #008080;">do</span> |
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atom t0 = time() |
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<span style="color: #004080;">atom</span> <span style="color: #000000;">t0</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">time</span><span style="color: #0000FF;">()</span> |
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for i=0 to 15 do |
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<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">0</span> <span style="color: #008080;">to</span> <span style="color: #000000;">15</span> <span style="color: #008080;">do</span> |
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switch t do |
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<span style="color: #008080;">switch</span> <span style="color: #000000;">t</span> <span style="color: #008080;">do</span> |
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case 1: res[i+1][2] = catalan1(i) |
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<span style="color: #008080;">case</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">:</span> <span style="color: #000000;">res</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">][</span><span style="color: #000000;">2</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">catalan1</span><span style="color: #0000FF;">(</span><span style="color: #000000;">i</span><span style="color: #0000FF;">)</span> |
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case 2: res[i+1][3] = catalan2(i) |
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<span style="color: #008080;">case</span> <span style="color: #000000;">2</span><span style="color: #0000FF;">:</span> <span style="color: #000000;">res</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">][</span><span style="color: #000000;">3</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">catalan2</span><span style="color: #0000FF;">(</span><span style="color: #000000;">i</span><span style="color: #0000FF;">)</span> |
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case 3: res[i+1][4] = catalan3(i) |
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<span style="color: #008080;">case</span> <span style="color: #000000;">3</span><span style="color: #0000FF;">:</span> <span style="color: #000000;">res</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">][</span><span style="color: #000000;">4</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">catalan3</span><span style="color: #0000FF;">(</span><span style="color: #000000;">i</span><span style="color: #0000FF;">)</span> |
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case 4: res[i+1][1] = i; printf(1,"%2d: %10d %10d %10d\n",res[i+1]) |
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<span style="color: #008080;">case</span> <span style="color: #000000;">4</span><span style="color: #0000FF;">:</span> <span style="color: #000000;">res</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">][</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">;</span> <span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"%2d: %10d %10d %10d\n"</span><span style="color: #0000FF;">,</span><span style="color: #000000;">res</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">])</span> |
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end switch |
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<span style="color: #008080;">end</span> <span style="color: #008080;">switch</span> |
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end for |
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<span style="color: #008080;">end</span> <span style="color: #008080;">for</span> |
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if t=4 then exit end if |
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<span style="color: #008080;">if</span> <span style="color: #000000;">t</span><span style="color: #0000FF;">=</span><span style="color: #000000;">4</span> <span style="color: #008080;">then</span> <span style="color: #008080;">exit</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span> |
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times[t] = elapsed(time()-t0) |
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<span style="color: #000000;">times</span><span style="color: #0000FF;">[</span><span style="color: #000000;">t</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">elapsed</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">time</span><span style="color: #0000FF;">()-</span><span style="color: #000000;">t0</span><span style="color: #0000FF;">)</span> |
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end for |
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<span style="color: #008080;">end</span> <span style="color: #008080;">for</span> |
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printf(1,"times:%8s %10s %10s\n",times)</lang> |
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<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"times:%8s %10s %10s\n"</span><span style="color: #0000FF;">,</span><span style="color: #000000;">times</span><span style="color: #0000FF;">)</span> |
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<!--</lang>--> |
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{{out}} |
{{out}} |
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<pre> |
<pre> |
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=== memoized recursive gmp version === |
=== memoized recursive gmp version === |
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{{libheader|Phix/mpfr}} |
{{libheader|Phix/mpfr}} |
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<lang Phix> |
<!--<lang Phix>--> |
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<span style="color: #008080;">include</span> <span style="color: #000000;">builtins</span><span style="color: #0000FF;">\</span><span style="color: #7060A8;">mpfr</span><span style="color: #0000FF;">.</span><span style="color: #000000;">e</span> |
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sequence c2cache = {} |
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<span style="color: #004080;">sequence</span> <span style="color: #000000;">c2cache</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{}</span> |
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function catalan2m(integer n) -- very fast! |
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<span style="color: #008080;">function</span> <span style="color: #000000;">catalan2m</span><span style="color: #0000FF;">(</span><span style="color: #004080;">integer</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">)</span> <span style="color: #000080;font-style:italic;">-- very fast!</span> |
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object r -- result (a [cached/shared] mpz) |
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<span style="color: #004080;">object</span> <span style="color: #000000;">r</span> <span style="color: #000080;font-style:italic;">-- result (a [cached/shared] mpz) |
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-- (nb: modifying result will mess up cache) |
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-- (nb: modifying result will mess up cache)</span> |
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if n<=0 then return mpz_init(1) end if |
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<span style="color: #008080;">if</span> <span style="color: #000000;">n</span><span style="color: #0000FF;"><=</span><span style="color: #000000;">0</span> <span style="color: #008080;">then</span> <span style="color: #008080;">return</span> <span style="color: #7060A8;">mpz_init</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span> |
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if n<=length(c2cache) then |
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<span style="color: #008080;">if</span> <span style="color: #000000;">n</span><span style="color: #0000FF;"><=</span><span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">c2cache</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">then</span> |
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r = c2cache[n] |
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<span style="color: #000000;">r</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">c2cache</span><span style="color: #0000FF;">[</span><span style="color: #000000;">n</span><span style="color: #0000FF;">]</span> |
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if r!=0 then return r end if |
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<span style="color: #008080;">if</span> <span style="color: #000000;">r</span><span style="color: #0000FF;">!=</span><span style="color: #000000;">0</span> <span style="color: #008080;">then</span> <span style="color: #008080;">return</span> <span style="color: #000000;">r</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span> |
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else |
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<span style="color: #008080;">else</span> |
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c2cache &= repeat(0,n-length(c2cache)) |
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<span style="color: #000000;">c2cache</span> <span style="color: #0000FF;">&=</span> <span style="color: #7060A8;">repeat</span><span style="color: #0000FF;">(</span><span style="color: #000000;">0</span><span style="color: #0000FF;">,</span><span style="color: #000000;">n</span><span style="color: #0000FF;">-</span><span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">c2cache</span><span style="color: #0000FF;">))</span> |
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end if |
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<span style="color: #008080;">end</span> <span style="color: #008080;">if</span> |
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r = mpz_init(0) |
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<span style="color: #000000;">r</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpz_init</span><span style="color: #0000FF;">(</span><span style="color: #000000;">0</span><span style="color: #0000FF;">)</span> |
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mpz t = mpz_init() |
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<span style="color: #7060A8;">mpz</span> <span style="color: #000000;">t</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpz_init</span><span style="color: #0000FF;">()</span> |
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for i=0 to n-1 do |
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<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">0</span> <span style="color: #008080;">to</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">-</span><span style="color: #000000;">1</span> <span style="color: #008080;">do</span> |
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mpz_mul(t,catalan2m(i),catalan2m(n-1-i)) |
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<span style="color: #7060A8;">mpz_mul</span><span style="color: #0000FF;">(</span><span style="color: #000000;">t</span><span style="color: #0000FF;">,</span><span style="color: #000000;">catalan2m</span><span style="color: #0000FF;">(</span><span style="color: #000000;">i</span><span style="color: #0000FF;">),</span><span style="color: #000000;">catalan2m</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">-</span><span style="color: #000000;">i</span><span style="color: #0000FF;">))</span> |
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mpz_add(r,r,t) |
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<span style="color: #7060A8;">mpz_add</span><span style="color: #0000FF;">(</span><span style="color: #000000;">r</span><span style="color: #0000FF;">,</span><span style="color: #000000;">r</span><span style="color: #0000FF;">,</span><span style="color: #000000;">t</span><span style="color: #0000FF;">)</span> |
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end for |
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<span style="color: #008080;">end</span> <span style="color: #008080;">for</span> |
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t = mpz_free(t) |
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<span style="color: #000000;">t</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpz_free</span><span style="color: #0000FF;">(</span><span style="color: #000000;">t</span><span style="color: #0000FF;">)</span> |
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c2cache[n] = r |
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<span style="color: #000000;">c2cache</span><span style="color: #0000FF;">[</span><span style="color: #000000;">n</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">r</span> |
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return r |
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<span style="color: #008080;">return</span> <span style="color: #000000;">r</span> |
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end function |
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<span style="color: #008080;">end</span> <span style="color: #008080;">function</span> |
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sequence s = {} |
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<span style="color: #004080;">sequence</span> <span style="color: #000000;">s</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{}</span> |
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for i=0 to 15 do s = append(s,mpz_get_str(catalan2m(i))) end for |
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<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">0</span> <span style="color: #008080;">to</span> <span style="color: #000000;">15</span> <span style="color: #008080;">do</span> <span style="color: #000000;">s</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">append</span><span style="color: #0000FF;">(</span><span style="color: #000000;">s</span><span style="color: #0000FF;">,</span><span style="color: #7060A8;">mpz_get_str</span><span style="color: #0000FF;">(</span><span style="color: #000000;">catalan2m</span><span style="color: #0000FF;">(</span><span style="color: #000000;">i</span><span style="color: #0000FF;">)))</span> <span style="color: #008080;">end</span> <span style="color: #008080;">for</span> |
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printf(1,"0..15: %s\n",join(s,",")) |
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<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"0..15: %s\n"</span><span style="color: #0000FF;">,</span><span style="color: #7060A8;">join</span><span style="color: #0000FF;">(</span><span style="color: #000000;">s</span><span style="color: #0000FF;">,</span><span style="color: #008000;">","</span><span style="color: #0000FF;">))</span> |
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printf(1,"100: %s\n",{mpz_get_str(catalan2m(100))})</lang> |
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<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"100: %s\n"</span><span style="color: #0000FF;">,{</span><span style="color: #7060A8;">mpz_get_str</span><span style="color: #0000FF;">(</span><span style="color: #000000;">catalan2m</span><span style="color: #0000FF;">(</span><span style="color: #000000;">100</span><span style="color: #0000FF;">))})</span> |
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<!--</lang>--> |
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{{out}} |
{{out}} |
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<pre> |
<pre> |