Last modified Tuesday, January 11, 2000 5:03 pm

The reverse sum (versum) sequence starting at 196

887,1675,7436,13783,52514,94039,187088,1067869,10755470,18211171,35322452,60744805,
111589511,227574622,454050344,897100798,1794102596,8746117567,16403234045,70446464506,
130992928913,450822227944,900544455998,1800098901007,8801197801088,17602285712176,
84724043932847,159547977975595,755127757721546,1400255515443103,4413700670963144,
8827391431036288,17653692772973576,85191620502609247,159482241005228405,
664304741147513356,1317620482294916822,3603815405135183953,7197630720180367016,
13305261530450734933,47248966933966985264,93507933867933969538,177104867844767940077,
947154635293536341848,1795298270686072793597,9749270977546801719568,
18408442064004592449047,92502871604050616929528,175095833209091234750057,
925153265399993573340628,1751196640799987135692157,9264161958699957602603728,
17537224026299926194218357,92918473189299188236491928,175837936477498486373973857,
934217310162393261013712428,1758434620324786522027424867,9442681822581660752291773438,
17786453745152322604573635887,96640091285774647759309104658,
182280281681549295517528109327,906182107397142240703710191608,
1712373124704184482497411473217,8836114272647029296571625205388,
17671139534403958504034349321776,85383533877444544434477942439447,
159876958854887988978955775977805,668656536414767878767414635656756,
1326313072829535757534829271313622,3589444802113893332894111974449853,
7178889593228875666877224058899706,13258878097456662332665448018788423,
45747659181913285659330927106673654,91385319354816681317562845302348408,
171869639709643252636224690693706727,899477035806065888888571598630674898,
1797953072701241777777132207161449896,8787394689723559555548553279865047867,
16474800379447118011108106559729985745,71233793175007298122189281057030833206,
131467596250025596244378551114170566423,456132667661181469687074071166866330554,
911166336322351940474038252333632562208,1713431572655604770948087405557266223327,
8946658200210652579438861471120017566498,17893315300422394267788614031240046132996,
87816479304635435956564863353640397472867,164643958609270772803130816807280794934745,
712083455691979390834439093880187654281206,1314265912473067781768877187859384208661423,
4555933937312655599557549065463126404285554,9111757983526301209015109021025263797681108,
17123625957151502418030218042061517695252227,
89348885628667526499233299462576693647884398,
178697760268335052998466598925153376306768796,
876565363941686582894131498175687238374565667,
1643130837774473154788262996461373387738131345,
7074449215608204801780891870975118165118444806,
13158897331226320592562872742059146230247889513,
44757771534490515617290699271561508443627774644,...

s=4:
no period detected 3,3,4,3,2,3,4,1,2,3,4,1,3,2,4,2,4,3,1,2,1,4,2,3,4,4,3,3,2,3,4,4,4,3,1,4,2,1,4,1,4,2,1,4, 1,4,3,4,1,4,1,4,1,4,1,4,3,2,3,2,3,4,1,4,4,3,1,4,2,1,2,3,2,4,3,2,4,3,1,2,3,2,4,3,2,4,3,1, 2,3,2,4,3,2,4,3,1,2,1,4,...



shaft utilization:

    1:  20%  ++++++++++++++++++++
    2:  22%  ++++++++++++++++++++++
    3:  26%  ++++++++++++++++++++++++++
    4:  32%  ++++++++++++++++++++++++++++++++

s=5:
no period detected 2,5,1,3,4,4,3,4,5,1,2,5,1,2,4,3,1,2,5,1,3,4,3,2,3,1,2,5,1,3,4,3,1,2,5,1,2,3,1,3,4,3,2,3, 2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,1,2,5,1,2,3,1,3,4,3,2,3,1,2,5,1,3,4,3,2,3,1,2,5, 1,3,4,3,2,3,1,2,5,1,3,4,...



shaft utilization:

    1:  19%  +++++++++++++++++++
    2:  27%  +++++++++++++++++++++++++++
    3:  33%  +++++++++++++++++++++++++++++++++
    4:  11%  +++++++++++
    5:  10%  ++++++++++

s=6:
no period detected 5,1,2,1,2,1,2,1,2,1,2,1,5,4,2,4,2,1,5,4,5,4,2,1,2,4,5,1,2,1,2,4,2,1,5,4,2,1,2,1,2,4,5,4, 5,4,5,4,5,4,5,4,5,4,5,4,5,4,5,4,5,4,5,4,2,1,5,4,2,1,2,1,2,4,5,4,2,1,5,4,5,4,2,1,2,4,5,1, 2,1,2,4,5,4,2,1,5,4,5,4,...



shaft utilization:

    1:  21%  +++++++++++++++++++++
    2:  25%  +++++++++++++++++++++++++
    3:   0%  
    4:  29%  +++++++++++++++++++++++++++++
    5:  25%  +++++++++++++++++++++++++
    6:   0%  

s=7:
no period detected 5,2,2,7,7,1,6,5,5,6,4,2,5,2,6,6,6,5,2,5,7,4,5,1,7,4,4,6,6,4,1,5,4,4,5,2,1,1,4,4,7,7,7,1, 5,4,2,7,6,7,3,5,7,4,3,5,2,4,3,1,4,4,2,4,6,4,3,5,4,4,3,6,2,5,3,5,7,6,7,5,7,5,5,5,3,6,3,2, 5,6,1,4,3,6,5,7,2,7,4,6,...



shaft utilization:

    1:   8%  ++++++++
    2:  12%  ++++++++++++
    3:   9%  +++++++++
    4:  20%  ++++++++++++++++++++
    5:  21%  +++++++++++++++++++++
    6:  15%  +++++++++++++++
    7:  15%  +++++++++++++++

s=8:
no period detected 7,3,4,7,2,7,8,5,6,3,4,5,7,6,8,6,4,7,5,2,1,8,6,7,8,8,7,3,2,7,8,8,8,7,5,4,6,1,8,5,8,2,5,8, 5,8,7,8,1,4,5,8,5,8,1,4,3,6,7,2,7,8,1,4,8,7,5,4,6,5,2,7,6,8,7,2,8,3,1,6,7,2,8,7,2,4,3,1, 6,7,2,4,3,6,4,3,1,6,1,4,...



shaft utilization:

    1:   9%  +++++++++
    2:  10%  ++++++++++
    3:   8%  ++++++++
    4:  12%  ++++++++++++
    5:  11%  +++++++++++
    6:  12%  ++++++++++++
    7:  18%  ++++++++++++++++++
    8:  20%  ++++++++++++++++++++

s=9:
period = 6; pre-periodic length = 0 [5,1,2,4,8,7] ...



shaft utilization:

    1:  17%  +++++++++++++++++
    2:  17%  +++++++++++++++++
    3:   0%  
    4:  17%  +++++++++++++++++
    5:  17%  +++++++++++++++++
    6:   0%  
    7:  16%  ++++++++++++++++
    8:  16%  ++++++++++++++++
    9:   0%  

s=10:
no period detected 7,5,6,3,4,9,8,9,10,1,2,5,1,2,4,8,6,7,5,6,3,4,8,7,8,6,7,5,6,3,4,8,6,7,5,6,2,3,6,3,4,8,7,8, 7,8,7,8,7,8,7,8,7,8,7,8,7,8,7,8,7,8,7,8,6,7,5,6,2,3,6,3,4,8,7,8,6,7,5,6,3,4,8,7,8,6,7,5, 6,3,4,8,7,8,6,7,5,6,3,4,...



shaft utilization:

    1:   2%  ++
    2:   4%  ++++
    3:  10%  ++++++++++
    4:   9%  +++++++++
    5:   9%  +++++++++
    6:  17%  +++++++++++++++++
    7:  23%  +++++++++++++++++++++++
    8:  23%  +++++++++++++++++++++++
    9:   2%  ++
   10:   1%  +

s=11:
period = 1; pre-periodic length = 2 7,3,[11] ...



shaft utilization:

    1:   0%  
    2:   0%  
    3:   1%  +
    4:   0%  
    5:   0%  
    6:   0%  
    7:   1%  +
    8:   0%  
    9:   0%  
   10:   0%  
   11:  98%  ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

s=12:
no period detected 11,7,8,7,2,7,8,1,2,7,8,1,11,10,8,10,8,7,5,10,5,4,2,7,8,4,11,7,2,7,8,4,8,7,5,4,2,1,8,1,8, 10,5,4,5,4,11,4,5,4,5,4,5,4,5,4,11,10,11,10,11,4,5,4,8,7,5,4,2,1,2,7,2,4,11,10,8,7,5,10, 11,10,8,7,2,4,11,1,2,7,2,4,11,10,8,7,5,10,5,4,...



shaft utilization:

    1:   6%  ++++++
    2:  11%  +++++++++++
    3:   0%  
    4:  18%  ++++++++++++++++++
    5:  14%  ++++++++++++++
    6:   0%  
    7:  15%  +++++++++++++++
    8:  14%  ++++++++++++++
    9:   0%  
   10:  11%  +++++++++++
   11:  11%  +++++++++++
   12:   0%  

s=13:
no period detected 3,11,13,3,7,10,5,10,11,4,9,4,7,2,7,9,13,5,3,2,2,10,5,4,4,9,7,1,1,7,2,6,7,3,4,7,9,13,4,10, 12,11,13,7,8,4,3,9,7,10,12,2,8,8,1,11,1,13,9,13,5,10,3,9,8,9,3,6,7,4,10,1,13,8,2,6,12,3, 10,11,3,12,10,6,12,11,7,2,10,9,1,9,2,13,13,2,7,2,8,7,...



shaft utilization:

    1:   6%  ++++++
    2:  10%  ++++++++++
    3:   9%  +++++++++
    4:   8%  ++++++++
    5:   4%  ++++
    6:   4%  ++++
    7:  13%  +++++++++++++
    8:   6%  ++++++
    9:  10%  ++++++++++
   10:  10%  ++++++++++
   11:   6%  ++++++
   12:   5%  +++++
   13:   9%  +++++++++

s=14:
no period detected 5,9,2,7,14,1,6,5,12,13,4,9,5,2,6,6,6,5,9,12,7,4,12,1,14,4,11,13,6,11,8,12,4,11,5,2,8,1,4, 11,14,14,7,8,5,4,9,14,13,14,3,12,7,4,3,12,9,4,3,8,11,4,9,4,6,11,3,12,4,11,10,13,2,12,3, 12,14,13,7,12,7,12,12,5,10,6,3,9,12,13,8,4,3,6,12,7,9,14,11,6,...



shaft utilization:

    1:   3%  +++
    2:   4%  ++++
    3:   7%  +++++++
    4:  12%  ++++++++++++
    5:   7%  +++++++
    6:   9%  +++++++++
    7:   7%  +++++++
    8:   5%  +++++
    9:   8%  ++++++++
   10:   2%  ++
   11:   8%  ++++++++
   12:  14%  ++++++++++++++
   13:   6%  ++++++
   14:   8%  ++++++++

s=15:
no period detected 2,10,11,13,14,4,8,4,5,1,2,10,11,7,14,13,11,7,5,1,8,4,8,7,8,1,2,10,11,13,14,13,11,7,5,1,2, 13,11,13,14,13,2,13,2,13,2,13,2,13,2,13,2,13,2,13,2,13,2,13,2,13,2,13,11,7,5,1,2,13,11, 13,14,13,2,13,11,7,5,1,8,4,8,7,8,1,2,10,11,13,14,13,2,13,11,7,5,1,8,4,...



shaft utilization:

    1:   8%  ++++++++
    2:  19%  +++++++++++++++++++
    3:   0%  
    4:   5%  +++++
    5:   6%  ++++++
    6:   0%  
    7:   8%  ++++++++
    8:   8%  ++++++++
    9:   0%  
   10:   4%  ++++
   11:  11%  +++++++++++
   12:   0%  
   13:  25%  +++++++++++++++++++++++++
   14:   6%  ++++++
   15:   0%  

s=16:
no period detected 7,11,12,7,2,7,16,13,14,3,4,5,7,14,8,14,4,15,13,10,1,8,14,15,16,16,15,11,10,15,8,16,8,15, 5,12,6,1,8,5,16,2,13,8,13,16,7,8,9,4,13,16,5,8,1,12,3,14,15,2,15,8,1,12,16,7,13,4,6,13, 10,7,6,8,7,2,8,11,1,6,7,10,16,15,2,4,3,9,6,15,2,4,3,14,12,3,1,6,9,4,...



shaft utilization:

    1:   6%  ++++++
    2:   6%  ++++++
    3:   5%  +++++
    4:   7%  +++++++
    5:   4%  ++++
    6:   6%  ++++++
    7:   9%  +++++++++
    8:  11%  +++++++++++
    9:   3%  +++
   10:   4%  ++++
   11:   3%  +++
   12:   5%  +++++
   13:   7%  +++++++
   14:   6%  ++++++
   15:   9%  +++++++++
   16:   9%  +++++++++

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e-mail: ralph@cs.arizona.edu

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