Pell numbers: Difference between revisions

</pre>

=={{header|Phix}}==

<!--<lang Phix>(phixonline)-->

<span style="color: #008080;">with</span> <span style="color: #008080;">javascript_semantics</span>

<span style="color: #004080;">sequence</span> <span style="color: #000000;">p</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">0</span><span style="color: #0000FF;">,</span><span style="color: #000000;">1</span><span style="color: #0000FF;">},</span>

<span style="color: #000000;">pl</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{</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: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">2</span> <span style="color: #008080;">to</span> <span style="color: #000000;">40</span> <span style="color: #008080;">do</span>

<span style="color: #000000;">p</span> <span style="color: #0000FF;">&=</span> <span style="color: #000000;">2</span><span style="color: #0000FF;">*</span><span style="color: #000000;">p</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">]+</span><span style="color: #000000;">p</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;">pl</span> <span style="color: #0000FF;">&=</span> <span style="color: #000000;">2</span><span style="color: #0000FF;">*</span><span style="color: #000000;">pl</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">]+</span><span style="color: #000000;">pl</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: #008080;">end</span> <span style="color: #008080;">for</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;">"First 20 Pell numbers: %s\n"</span><span style="color: #0000FF;">,{</span><span style="color: #7060A8;">join_by</span><span style="color: #0000FF;">(</span><span style="color: #000000;">p</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">..</span><span style="color: #000000;">20</span><span style="color: #0000FF;">],</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #000000;">20</span><span style="color: #0000FF;">,</span><span style="color: #008000;">" "</span><span style="color: #0000FF;">,</span><span style="color: #000000;">fmt</span><span style="color: #0000FF;">:=</span><span style="color: #008000;">"%d"</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;">"First 20 Pell-Lucas: %s\n"</span><span style="color: #0000FF;">,{</span><span style="color: #7060A8;">join_by</span><span style="color: #0000FF;">(</span><span style="color: #000000;">pl</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">..</span><span style="color: #000000;">20</span><span style="color: #0000FF;">],</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #000000;">20</span><span style="color: #0000FF;">,</span><span style="color: #008000;">" "</span><span style="color: #0000FF;">,</span><span style="color: #000000;">fmt</span><span style="color: #0000FF;">:=</span><span style="color: #008000;">"%d"</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;">"First 20 rational approximations of sqrt(2) (%.16f):\n"</span><span style="color: #0000FF;">,{</span><span style="color: #7060A8;">sqrt</span><span style="color: #0000FF;">(</span><span style="color: #000000;">2</span><span style="color: #0000FF;">)})</span>

<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">2</span> <span style="color: #008080;">to</span> <span style="color: #000000;">21</span> <span style="color: #008080;">do</span>

<span style="color: #004080;">integer</span> <span style="color: #000000;">n</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">pl</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">]/</span><span style="color: #000000;">2</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">d</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">p</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;">"%d/%d ~= %.16g\n"</span><span style="color: #0000FF;">,</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">n</span><span style="color: #0000FF;">,</span><span style="color: #000000;">d</span><span style="color: #0000FF;">,</span><span style="color: #000000;">n</span><span style="color: #0000FF;">/</span><span style="color: #000000;">d</span><span style="color: #0000FF;">})</span>

<span style="color: #008080;">end</span> <span style="color: #008080;">for</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;">"\nFirst 20 Pell primes:\n"</span><span style="color: #0000FF;">)</span>

<span style="color: #008080;">include</span> <span style="color: #004080;">mpfr</span><span style="color: #0000FF;">.</span><span style="color: #000000;">e</span>

<span style="color: #004080;">mpz</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">p0</span><span style="color: #0000FF;">,</span><span style="color: #000000;">p1</span><span style="color: #0000FF;">,</span><span style="color: #000000;">p2</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpz_inits</span><span style="color: #0000FF;">(</span><span style="color: #000000;">3</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">0</span><span style="color: #0000FF;">,</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #000000;">0</span><span style="color: #0000FF;">})</span>

<span style="color: #004080;">sequence</span> <span style="color: #000000;">ppdx</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{}</span>

<span style="color: #004080;">integer</span> <span style="color: #000000;">pdx</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">2</span>

<span style="color: #008080;">while</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">ppdx</span><span style="color: #0000FF;">)<</span><span style="color: #000000;">20</span> <span style="color: #008080;">do</span>

<span style="color: #7060A8;">mpz_mul_si</span><span style="color: #0000FF;">(</span><span style="color: #000000;">p2</span><span style="color: #0000FF;">,</span><span style="color: #000000;">p1</span><span style="color: #0000FF;">,</span><span style="color: #000000;">2</span><span style="color: #0000FF;">)</span>

<span style="color: #7060A8;">mpz_add</span><span style="color: #0000FF;">(</span><span style="color: #000000;">p2</span><span style="color: #0000FF;">,</span><span style="color: #000000;">p2</span><span style="color: #0000FF;">,</span><span style="color: #000000;">p0</span><span style="color: #0000FF;">)</span>

<span style="color: #008080;">if</span> <span style="color: #7060A8;">is_prime</span><span style="color: #0000FF;">(</span><span style="color: #000000;">pdx</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">and</span> <span style="color: #7060A8;">mpz_prime</span><span style="color: #0000FF;">(</span><span style="color: #000000;">p2</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">then</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;">"%s\n"</span><span style="color: #0000FF;">,</span><span style="color: #7060A8;">mpz_get_short_str</span><span style="color: #0000FF;">(</span><span style="color: #000000;">p2</span><span style="color: #0000FF;">))</span>

<span style="color: #000000;">ppdx</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">append</span><span style="color: #0000FF;">(</span><span style="color: #000000;">ppdx</span><span style="color: #0000FF;">,</span><span style="color: #7060A8;">sprintf</span><span style="color: #0000FF;">(</span><span style="color: #008000;">"%d"</span><span style="color: #0000FF;">,</span><span style="color: #000000;">pdx</span><span style="color: #0000FF;">))</span>

<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>

<span style="color: #000000;">pdx</span> <span style="color: #0000FF;">+=</span> <span style="color: #000000;">1</span>

<span style="color: #7060A8;">mpz_set</span><span style="color: #0000FF;">(</span><span style="color: #000000;">p0</span><span style="color: #0000FF;">,</span><span style="color: #000000;">p1</span><span style="color: #0000FF;">)</span>

<span style="color: #7060A8;">mpz_set</span><span style="color: #0000FF;">(</span><span style="color: #000000;">p1</span><span style="color: #0000FF;">,</span><span style="color: #000000;">p2</span><span style="color: #0000FF;">)</span>

<span style="color: #008080;">end</span> <span style="color: #008080;">while</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;">"\nIndices of first 20 Pell primes: %s\n"</span><span style="color: #0000FF;">,</span><span style="color: #7060A8;">join</span><span style="color: #0000FF;">(</span><span style="color: #000000;">ppdx</span><span style="color: #0000FF;">,</span><span style="color: #008000;">" "</span><span style="color: #0000FF;">))</span>

<span style="color: #004080;">sequence</span> <span style="color: #000000;">nsw</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{}</span>

<span style="color: #008080;">for</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">20</span> <span style="color: #008080;">do</span> <span style="color: #000000;">nsw</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">append</span><span style="color: #0000FF;">(</span><span style="color: #000000;">nsw</span><span style="color: #0000FF;">,</span><span style="color: #7060A8;">sprintf</span><span style="color: #0000FF;">(</span><span style="color: #008000;">"%d"</span><span style="color: #0000FF;">,</span><span style="color: #000000;">p</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;">p</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: #008080;">end</span> <span style="color: #008080;">for</span>

<span style="color: #000000;">nsw</span><span style="color: #0000FF;">[</span><span style="color: #000000;">8</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: #0000FF;">{</span><span style="color: #008000;">"..."</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;">"\nFirst 20 Newman-Shank-Williams numbers: %s\n"</span><span style="color: #0000FF;">,{</span><span style="color: #7060A8;">join</span><span style="color: #0000FF;">(</span><span style="color: #000000;">nsw</span><span style="color: #0000FF;">,</span><span style="color: #008000;">" "</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;">"\nFirst 20 near isosceles right triangles:\n"</span><span style="color: #0000FF;">)</span>

<span style="color: #004080;">atom</span> <span style="color: #000000;">i0</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">i1</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">i2</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">t</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> <span style="color: #000000;">2</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">found</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">0</span>

<span style="color: #008080;">while</span> <span style="color: #000000;">found</span><span style="color: #0000FF;"><</span><span style="color: #000000;">20</span> <span style="color: #008080;">do</span>

<span style="color: #000000;">i2</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">i1</span><span style="color: #0000FF;">*</span><span style="color: #000000;">2</span><span style="color: #0000FF;">+</span><span style="color: #000000;">i0</span>

<span style="color: #008080;">if</span> <span style="color: #7060A8;">odd</span><span style="color: #0000FF;">(</span><span style="color: #000000;">i</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">then</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;">"[%d, %d, %d]\n"</span><span style="color: #0000FF;">,</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">t</span><span style="color: #0000FF;">,</span><span style="color: #000000;">t</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #000000;">i2</span><span style="color: #0000FF;">})</span>

<span style="color: #000000;">found</span> <span style="color: #0000FF;">+=</span> <span style="color: #000000;">1</span>

<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>

<span style="color: #000000;">t</span> <span style="color: #0000FF;">+=</span> <span style="color: #000000;">i2</span>

<span style="color: #0000FF;">{</span><span style="color: #000000;">i0</span><span style="color: #0000FF;">,</span><span style="color: #000000;">i1</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">i1</span><span style="color: #0000FF;">,</span><span style="color: #000000;">i2</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: #008080;">end</span> <span style="color: #008080;">while</span>

<!--</lang>-->

{{out}}

<pre>

First 20 Pell numbers: 0 1 2 5 12 29 70 169 408 985 2378 5741 13860 33461 80782 195025 470832 1136689 2744210 6625109

First 20 Pell-Lucas: 2 2 6 14 34 82 198 478 1154 2786 6726 16238 39202 94642 228486 551614 1331714 3215042 7761798 18738638

First 20 rational approximations of sqrt(2) (1.4142135623730951):

1/1 ~= 1

3/2 ~= 1.5

7/5 ~= 1.4

17/12 ~= 1.416666666666667

41/29 ~= 1.413793103448276

99/70 ~= 1.414285714285714

239/169 ~= 1.414201183431953

577/408 ~= 1.41421568627451

1393/985 ~= 1.414213197969543

3363/2378 ~= 1.41421362489487

8119/5741 ~= 1.414213551646055

19601/13860 ~= 1.414213564213564

47321/33461 ~= 1.41421356205732

114243/80782 ~= 1.414213562427273

275807/195025 ~= 1.414213562363799

665857/470832 ~= 1.41421356237469

1607521/1136689 ~= 1.414213562372821

3880899/2744210 ~= 1.414213562373142

9369319/6625109 ~= 1.414213562373087

22619537/15994428 ~= 1.414213562373097

First 20 Pell primes:

2

5

29

5741

33461

44560482149

1746860020068409

68480406462161287469

13558774610046711780701

4125636888562548868221559797461449

4760981394323203445293052612223893281

161733217200188571081311986634082331709

29647935552727996719...32712780937764687561 (64 digits)

67741382025708508432...87756429585398537901 (69 digits)

45562852543334487715...91808671945330235841 (73 digits)

54971607658948646301...49971207793263738989 (200 digits)

14030291214037674827...9886357032327271649 (356 digits)

24348043146521993819...3590369653438042689 (466 digits)

34643489561492944482...62121115635939797709 (498 digits)

32074710952523740376...14678081652481281009 (521 digits)

Indices of first 20 Pell primes: 2 3 5 11 13 29 41 53 59 89 97 101 167 181 191 523 929 1217 1301 1361

First 20 Newman-Shank-Williams numbers: 1 7 41 239 1393 8119 47321 ... 72722761475561 423859315570607

First 20 near isosceles right triangles:

[3, 4, 5]

[20, 21, 29]

[119, 120, 169]

[696, 697, 985]

[4059, 4060, 5741]

[23660, 23661, 33461]

[137903, 137904, 195025]

[803760, 803761, 1136689]

[4684659, 4684660, 6625109]

[27304196, 27304197, 38613965]

[159140519, 159140520, 225058681]

[927538920, 927538921, 1311738121]

[5406093003, 5406093004, 7645370045]

[31509019100, 31509019101, 44560482149]

[183648021599, 183648021600, 259717522849]

[1070379110496, 1070379110497, 1513744654945]

[6238626641379, 6238626641380, 8822750406821]

[36361380737780, 36361380737781, 51422757785981]

[211929657785303, 211929657785304, 299713796309065]

[1235216565974040, 1235216565974041, 1746860020068409]