Pi chunked to form a (self-describing) succession

of odd and even terms

SeqFan mailing list [June 22nd, 2010]

« a(0) = 3; for n>0, break up decimal expansion of Pi into chunks that describe alternatively runs of odd and even parity terms »:

S = 3, 1, 41, 592, 65, 3, 5, 89, 7, 9, 3, 23, 8462643, 3, 83, 27, 9, 5028841, 9, 7, 1, 69, 3, 9, 9, 3, 7, 5, 105, 8209, 7, 49, 445, 9, 2307, 81, 64062862089, 9, 862803, 4825, 3, 421, 1, 7067, 9, 8, 2, ...

Example (‘o’ is for ‘odd term’ and ‘E’ for ‘Even term’; changes of parity are marked in grey):

S=3,1,41,592,65,3,5,89,7,9,3,23,8462643,3,83,27,9,5028841,9,7,1,69,3,9, 9,3,7,5,105,8209,7,49,445,9,2307,81,64062862089,9,862803,4825,3,4211,7067,9,8,2,...

o o o   E   o o o  o o o o  o    o    o  o  o o    o    o o o  o o o  o o o o  o    o  o  o  o  o   o   o       o     o    o     o  o  o  o  o  o E E ...

3     1                                                        41                <---- this is S again ---->                                         592

... which reads: "In this seq there are 3 odd terms, followed by 1 even term, then 41 odd terms, then 592 even terms, etc.

Maximilian Hasler was quick to compute the next even chunks; we thus have now for S (odd chunks italicized):

S = 3, 1, 41, 592, 65, 3, 5, 89, 7, 9, 3, 23, 8462643, 3, 83, 27, 9, 5028841, 9, 7, 1, 69, 3, 9,  9, 3, 7, 5, 105, 8209, 7, 49, 445, 9, 2307, 81, 64062862089, 9, 862803, 4825, 3, 4211, 7067, 9, 8, 2, 14, 80, 8, 6, 5132, 8, 2, 30, 6, 6, 4, 70, 938, 4, 4, 60, 9550, 58, 2, 2, 3172, 535940, 8, 12, 8, 4, 8, 11174, 50, 2, 8, 4, 10, 2, 70, 1938, 52, 110, 55596, 4, 4, 6, 2, 2, 94, 8, 954, 930, 38, 196, 4, 4, 2, 8, 8, 10, 9756, 6, 59334, 4, 6, 12, 8, 4, 756, 4, 8, 2, 3378, 6, 78, 316, 52, 7120, 190, 914, 56, 4, 8, 56, 6, 92, 34, 60, 34, 8, 6, 10, 4, 54, 32, 6, 6, 4, 8, 2, 1339360, 72, 60, 2, 4, 914, 12, 7372, 4, 58, 700, 6, 60, 6, 31558, 8, 174, 8, 8, 1520, 920, 96, 2, 8, 2, 92, 540, 9171536, 4, 36, 78, 92, 590, 36o0, 11330, 530, 54, 8, 8, 20, 4, 6, 6, 52, 138, 4, 14, 6, 95194, 151160, 94, 330, 572, 70, 36, 57595919530, 92, 18, 6, 11738, 1932, 6, 1179310, 5118, 54, 80, 74, 4, 6, 2, 37996, 2, 74, 956, 73518, 8, 5752, 72, 4, 8, 912, 2, 7938, 18, 30, 1194, 912, 98, 336, 7336, 2, 4, 40, 6, 56, 6, 4, 30, 8, 60, 2, 1394, 94, 6, 3952, 2, 4, 737190, 70, 2, 1798, 60, 94, 370, 2, 770, 5392, 17176, 2, 93176, 752, 38, 4, 6, 74, 8, 18, 4, 6, 76, 6, 940, 5132oo0, 56, 8, 12, 714, 52, 6, 3560, 8, 2, 778, 577134, 2, 75778, 960, 91736, 37178, 72, 14, 6, 8, 4, 40, 90, 12, 2, 4, 9534, 30, 14, 6, 54, 958, 53710, 50, 792, 2, 796, 8, 92, 58, 92, 354, 20, 19956, 112, 12, 90, 2, 1960, 8, 6, 40, 34, 4, 18, 1598, 136, 2, 9774, 77130, 9960, 518, 70, 72, 1134, 9999998, 372, 9780, 4, 99510, 59731732, 8, 160, 96, 318, 5950, 2, 4, 4, 594, 5534, 6, 90, 8, 30, 2, 6, 4, 2, 52, 2, 30, 8, 2, 5334, 4, 6, 8, 50, 352, 6, 193118, 8, 1710, 10o0, 31378, 38, 752, 8, 8, 6, 58, 753320, 8, 38, 14, 20, 6, 171776, 6, 914, 730, 3598, 2, 534, 90, 4, 2, 8, 7554, 6, 8, 73115956, 2, 8, 6, 38, 8, 2, 35378, 759375195778, 18, 57780, 532, 1712, 2, 6, 80, 6, 6, 1300, 192, 78, 76, 6, 1119590, 92, 16, 4, 20, 198, 9380, 952, 5720, 10, 6, 54, 8, 58, 6, 32, 78, 8, 6, 5936, 15338, 18, 2, 796, 8,  2, 30, 30, 19520, 3530, 18, 52, 96, 8, 9957736, 2, 2, 5994, 138, 912, 4, 972, 17752, 8, 34, 79131515574, 8, 572, 4, 2, 4, 54, 150, 6, 95950, 8, 2, 9533116, 8, 6, 172, 78, 558, 8, 90, 750, 98, 38, 1754, 6, 374, 6, 4, 9393192, 550, 60, 4o0, 92, 770, 16, 7113900, 98, 4, 8, 8, 2, 40, 12, 8, 58, 36, 160, 356, 370, 76, 60, 10, 4, 710, 18, 194, 2, 955596, 198, 94, 6, 76, 78, 374, 4, 94, 4, 8, 2, 55379774, 72, 6, 8, 4, 710, 40, 4, 7534, 6, 4, 6, 20, 80, 4, 6, 6, 8, 4, 2, 590, 6, 94, 912, 933136, 770, 2, 8, 98, 9152, 10, 4, 752, 16, 20, 56, 96, 60, 2, 40, 580, 38, 150, 1935112, 5338, 2, 4300, 3558, 76, 40, 2, 4, 74, 96, 4, 732, 6, 3914, 1992, 72, 60, 4, 2, 6, 992, 2, 796, 78, 2, 354, 78, 163600, 934, 172, 16, 4, 12, 1992, 4, 58, 6, 3150, 30, 2, 8, 6, 18, 2, 974, 55570, 6, 74, 9, 83, 8505, 49, 45, 885, 869, 269, 9, 5, 6909, 27, 2107, 9, 7, 509, 3029, 5, 5, 3, 21, 1, 65, 3, 449, 87, 2027, 5, 5, 9, 6023, 6480665, 49, 9, 1, 1, 9, 881, 83, 47, 9, 7, 7, 5, 3, 5, 663, 69, 807, 4265, 425, 27, 8625, 5, 1, 81, 841, 7, 5, 7, 467, 28909, 7, 7, 7, 72, 7938000, ...

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Note:

S is not Neil Sloane’s A136517:

« a(0) = 3; for n>0, break up decimal expansion of Pi into chunks of increasing lengths; leading zeros are not printed. »

S’ = 3, 1, 41, 592, 6535, 89793, 238462, 6433832, 79502884, 197169399, ...

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Jeremy Gardiner:

Eric,

Very nice sequence idea!

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Thanks to all!

Best,

É.