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

The triangular numbers.

1,3,6,10,15,21,28,36,45,55,66,78,91,105,120,136,153,171,190,210,231,253,276,300,325,351,
378,406,435,465,496,528,561,595,630,666,703,741,780,820,861,903,946,990,1035,1081,1128,
1176,1225,1275,1326,1378,1431,1485,1540,1596,1653,1711,1770,1830,1891,1953,2016,2080,
2145,2211,2278,2346,2415,2485,2556,2628,2701,2775,2850,2926,3003,3081,3160,3240,3321,
3403,3486,3570,3655,3741,3828,3916,4005,4095,4186,4278,4371,4465,4560,4656,4753,4851,
4950,5050,...

s=4:
period = 8; pre-periodic length = 0 [1,3,2,2,3,1,4,4] ...



shaft utilization:

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

s=5:
period = 5; pre-periodic length = 0 [1,3,1,5,5] ...



shaft utilization:

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

s=6:
period = 12; pre-periodic length = 0 [1,3,6,4,3,3,4,6,3,1,6,6] ...



shaft utilization:

    1:  17%  +++++++++++++++++
    2:   0%  
    3:  33%  +++++++++++++++++++++++++++++++++
    4:  17%  +++++++++++++++++
    5:   0%  
    6:  33%  +++++++++++++++++++++++++++++++++

s=7:
period = 7; pre-periodic length = 0 [1,3,6,3,1,7,7] ...



shaft utilization:

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

s=8:
period = 16; pre-periodic length = 0 [1,3,6,2,7,5,4,4,5,7,2,6,3,1,8,8] ...



shaft utilization:

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

s=9:
period = 9; pre-periodic length = 0 [1,3,6,1,6,3,1,9,9] ...



shaft utilization:

    1:  34%  ++++++++++++++++++++++++++++++++++
    2:   0%  
    3:  22%  ++++++++++++++++++++++
    4:   0%  
    5:   0%  
    6:  22%  ++++++++++++++++++++++
    7:   0%  
    8:   0%  
    9:  22%  ++++++++++++++++++++++

s=10:
period = 20; pre-periodic length = 0 [1,3,6,10,5,1,8,6,5,5,6,8,1,5,10,6,3,1,10,10] ...



shaft utilization:

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

s=11:
period = 11; pre-periodic length = 0 [1,3,6,10,4,10,6,3,1,11,11] ...



shaft utilization:

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

s=12:
period = 24; pre-periodic length = 0 [1,3,6,10,3,9,4,12,9,7,6,6,7,9,12,4,9,3,10,6,3,1,12,12] ...



shaft utilization:

    1:   9%  +++++++++
    2:   0%  
    3:  17%  +++++++++++++++++
    4:   8%  ++++++++
    5:   0%  
    6:  17%  +++++++++++++++++
    7:   8%  ++++++++
    8:   0%  
    9:  16%  ++++++++++++++++
   10:   9%  +++++++++
   11:   0%  
   12:  16%  ++++++++++++++++

s=13:
period = 13; pre-periodic length = 0 [1,3,6,10,2,8,2,10,6,3,1,13,13] ...



shaft utilization:

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

s=14:
period = 28; pre-periodic length = 0 [1,3,6,10,1,7,14,8,3,13,10,8,7,7,8,10,13,3,8,14,7,1,10,6,3,1,14,14] ...



shaft utilization:

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

s=15:
period = 15; pre-periodic length = 0 [1,3,6,10,15,6,13,6,15,10,6,3,1,15,15] ...



shaft utilization:

    1:  13%  +++++++++++++
    2:   0%  
    3:  13%  +++++++++++++
    4:   0%  
    5:   0%  
    6:  27%  +++++++++++++++++++++++++++
    7:   0%  
    8:   0%  
    9:   0%  
   10:  14%  ++++++++++++++
   11:   0%  
   12:   0%  
   13:   7%  +++++++
   14:   0%  
   15:  26%  ++++++++++++++++++++++++++

s=16:
period = 32; pre-periodic length = 0 [1,3,6,10,15,5,12,4,13,7,2,14,11,9,8,8,9,11,14,2,7,13,4,12,5,15,10,6,3,1,16,16] ...



shaft utilization:

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

back

e-mail: ralph@cs.arizona.edu

home