title: Fastest method to define whether a number is a triangular number
body: <p>A triangular number is the sum of the n natural numbers from 1 to n. What is the fastest method to find whether a given positive integer number is a triangular one? 
  
  
  <p>Here is a cut of the first 1200th up to 1300th triangular numbers, you can easily see a bit-pattern here (if not, try to zoom out):</p>
  
  <pre><code>(720600, '10101111111011011000')
  (721801, '10110000001110001001')
  (723003, '10110000100000111011')
  (724206, '10110000110011101110')
  (725410, '10110001000110100010')
  (726615, '10110001011001010111')
  (727821, '10110001101100001101')
  (729028, '10110001111111000100')
  (730236, '10110010010001111100')
  (731445, '10110010100100110101')
  (732655, '10110010110111101111')
  (733866, '10110011001010101010')
  (735078, '10110011011101100110')
  (736291, '10110011110000100011')
  (737505, '10110100000011100001')
  (738720, '10110100010110100000')
  (739936, '10110100101001100000')
  (741153, '10110100111100100001')
  (742371, '10110101001111100011')
  (743590, '10110101100010100110')
  (744810, '10110101110101101010')
  (746031, '10110110001000101111')
  (747253, '10110110011011110101')
  (748476, '10110110101110111100')
  (749700, '10110111000010000100')
  (750925, '10110111010101001101')
  (752151, '10110111101000010111')
  (753378, '10110111111011100010')
  (754606, '10111000001110101110')
  (755835, '10111000100001111011')
  (757065, '10111000110101001001')
  (758296, '10111001001000011000')
  (759528, '10111001011011101000')
  (760761, '10111001101110111001')
  (761995, '10111010000010001011')
  (763230, '10111010010101011110')
  (764466, '10111010101000110010')
  (765703, '10111010111100000111')
  (766941, '10111011001111011101')
  (768180, '10111011100010110100')
  (769420, '10111011110110001100')
  (770661, '10111100001001100101')
  (771903, '10111100011100111111')
  (773146, '10111100110000011010')
  (774390, '10111101000011110110')
  (775635, '10111101010111010011')
  (776881, '10111101101010110001')
  (778128, '10111101111110010000')
  (779376, '10111110010001110000')
  (780625, '10111110100101010001')
  (781875, '10111110111000110011')
  (783126, '10111111001100010110')
  (784378, '10111111011111111010')
  (785631, '10111111110011011111')
  (786885, '11000000000111000101')
  (788140, '11000000011010101100')
  (789396, '11000000101110010100')
  (790653, '11000001000001111101')
  (791911, '11000001010101100111')
  (793170, '11000001101001010010')
  (794430, '11000001111100111110')
  (795691, '11000010010000101011')
  (796953, '11000010100100011001')
  (798216, '11000010111000001000')
  (799480, '11000011001011111000')
  (800745, '11000011011111101001')
  (802011, '11000011110011011011')
  (803278, '11000100000111001110')
  (804546, '11000100011011000010')
  (805815, '11000100101110110111')
  (807085, '11000101000010101101')
  (808356, '11000101010110100100')
  (809628, '11000101101010011100')
  (810901, '11000101111110010101')
  (812175, '11000110010010001111')
  (813450, '11000110100110001010')
  (814726, '11000110111010000110')
  (816003, '11000111001110000011')
  (817281, '11000111100010000001')
  (818560, '11000111110110000000')
  (819840, '11001000001010000000')
  (821121, '11001000011110000001')
  (822403, '11001000110010000011')
  (823686, '11001001000110000110')
  (824970, '11001001011010001010')
  (826255, '11001001101110001111')
  (827541, '11001010000010010101')
  (828828, '11001010010110011100')
  (830116, '11001010101010100100')
  (831405, '11001010111110101101')
  (832695, '11001011010010110111')
  (833986, '11001011100111000010')
  (835278, '11001011111011001110')
  (836571, '11001100001111011011')
  (837865, '11001100100011101001')
  (839160, '11001100110111111000')
  (840456, '11001101001100001000')
  (841753, '11001101100000011001')
  (843051, '11001101110100101011')
  (844350, '11001110001000111110')
  </code></pre>
  
  <p>For example, can you also see a rotated normal distribution curve, represented by zeros between 807085 and 831405? This pattern repeats itself regularly.--></p>