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

The pentagonal numbers.

1,5,12,22,35,51,70,92,117,145,176,210,247,287,330,376,425,477,532,590,651,715,782,852,
925,1001,1080,1162,1247,1335,1426,1520,1617,1717,1820,1926,2035,2147,2262,2380,2501,2625,
2752,2882,3015,3151,3290,3432,3577,3725,3876,4030,4187,4347,4510,4676,4845,5017,5192,
5370,5551,5735,5922,6112,6305,6501,6700,6902,7107,7315,7526,7740,7957,8177,8400,8626,
8855,9087,9322,9560,9801,10045,10292,10542,10795,11051,11310,11572,11837,12105,12376,
12650,12927,13207,13490,13776,14065,14357,14652,14950,...

s=4:

period = 8; pre-periodic length = 0

[1,1,4,2,3,3,2,4] ...

shaft utilization:

    1:  26%  ++++++++++++++++++++++++++
    2:  25%  +++++++++++++++++++++++++
    3:  24%  ++++++++++++++++++++++++
    4:  25%  +++++++++++++++++++++++++

s=5:

period = 5; pre-periodic length = 0

[1,5,2,2,5] ...

shaft utilization:

    1:  20%  ++++++++++++++++++++
    2:  40%  ++++++++++++++++++++++++++++++++++++++++
    3:   0%  
    4:   0%  
    5:  40%  ++++++++++++++++++++++++++++++++++++++++

s=6:

period = 12; pre-periodic length = 0

[1,5,6,4,5,3,4,2,3,1,2,6] ...

shaft utilization:

    1:  17%  +++++++++++++++++
    2:  16%  ++++++++++++++++
    3:  16%  ++++++++++++++++
    4:  17%  +++++++++++++++++
    5:  17%  +++++++++++++++++
    6:  17%  +++++++++++++++++

s=7:

period = 7; pre-periodic length = 0

[1,5,5,1,7,2,7] ...

shaft utilization:

    1:  29%  +++++++++++++++++++++++++++++
    2:  14%  ++++++++++++++
    3:   0%  
    4:   0%  
    5:  29%  +++++++++++++++++++++++++++++
    6:   0%  
    7:  28%  ++++++++++++++++++++++++++++

s=8:

period = 16; pre-periodic length = 0

[1,5,4,6,3,3,6,4,5,1,8,2,7,7,2,8] ...

shaft utilization:

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

s=9:

period = 9; pre-periodic length = 0

[1,5,3,4,8,6,7,2,9] ...

shaft utilization:

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

s=10:

period = 20; pre-periodic length = 0

[1,5,2,2,5,1,10,2,7,5,6,10,7,7,10,6,5,7,2,10] ...

shaft utilization:

    1:  10%  ++++++++++
    2:  20%  ++++++++++++++++++++
    3:   0%  
    4:   0%  
    5:  20%  ++++++++++++++++++++
    6:  10%  ++++++++++
    7:  20%  ++++++++++++++++++++
    8:   0%  
    9:   0%  
   10:  20%  ++++++++++++++++++++

s=11:

period = 11; pre-periodic length = 0

[1,5,1,11,2,7,4,4,7,2,11] ...

shaft utilization:

    1:  19%  +++++++++++++++++++
    2:  18%  ++++++++++++++++++
    3:   0%  
    4:  18%  ++++++++++++++++++
    5:   9%  +++++++++
    6:   0%  
    7:  18%  ++++++++++++++++++
    8:   0%  
    9:   0%  
   10:   0%  
   11:  18%  ++++++++++++++++++

s=12:

period = 24; pre-periodic length = 0

[1,5,12,10,11,3,10,8,9,1,8,6,7,11,6,4,5,9,4,2,3,7,2,12] ...

shaft utilization:

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

s=13:

period = 13; pre-periodic length = 0

[1,5,12,9,9,12,5,1,13,2,7,2,13] ...

shaft utilization:

    1:  16%  ++++++++++++++++
    2:  14%  ++++++++++++++
    3:   0%  
    4:   0%  
    5:  16%  ++++++++++++++++
    6:   0%  
    7:   7%  +++++++
    8:   0%  
    9:  16%  ++++++++++++++++
   10:   0%  
   11:   0%  
   12:  16%  ++++++++++++++++
   13:  15%  +++++++++++++++

s=14:

period = 28; pre-periodic length = 0

[1,5,12,8,7,9,14,8,5,5,8,14,9,7,8,12,5,1,14,2,7,1,12,12,1,7,2,14] ...

shaft utilization:

    1:  13%  +++++++++++++
    2:   6%  ++++++
    3:   0%  
    4:   0%  
    5:  15%  +++++++++++++++
    6:   0%  
    7:  14%  ++++++++++++++
    8:  16%  ++++++++++++++++
    9:   8%  ++++++++
   10:   0%  
   11:   0%  
   12:  14%  ++++++++++++++
   13:   0%  
   14:  14%  ++++++++++++++

s=15:

period = 15; pre-periodic length = 0

[1,5,12,7,5,6,10,2,12,10,11,15,7,2,15] ...

shaft utilization:

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

s=16:

period = 32; pre-periodic length = 0

[1,5,12,6,3,3,6,12,5,1,16,2,7,15,10,8,9,13,4,14,11,11,14,4,13,9,8,10,15,7,2,16] ...

shaft utilization:

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

e-mail: ralph@cs.arizona.edu

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