Counting  1+1+1+1+1+...  1, 2, 3, 4, 5,...  

Triangular  1+2+3+4+5+...  1, 3, 6, 10, 15,...  
Square  1+3+5+7+9+...  1, 4, 9, 16, 25,...  
Pentagonal  1+4+7+10+13+...  1, 5, 12, 22, 35,...  
Hexagonal  1+5+9+13+17+...  1, 6, 15, 28, 45,...  
Heptagonal  1+6+11+16+21+...  1, 7, 18, 34, 55,...  
Octagonal  1+7+13+19+25+...  1, 8, 21, 40, 65,...  
Fibonacci  1+1,1+2,2+3,3+5,5+8,...  2, 3, 5, 8, 13,...  
Tetrahedral  1+3+6+10+15+...  1, 4, 10, 20, 35,...  (1/6)n(n+1)(n+2) 
Square Pyramid  1^{2}+2^{2}+3^{2}+4^{2}+5^{2}+...  1, 5, 14, 30, 55,...  (1/6)n(n+1)(2n+1) 
Squares  1^{2}, 2^{2}, 3^{2}, 4^{2}, 5^{2},...  1, 4, 9, 16, 25,...  
Cubes  1^{3}, 2^{3}, 3^{3}, 4^{3}, 5^{3},...  1, 8, 27, 64, 125,...  
4D Pantatope  1+4+10+20+35+...  1, 5, 15, 35, 70,,... 
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12,...
Every number is higher than the previous by the sum of 1, that is, 1 + 1 = 2, 4 + 1 = 5, etc., like this:
1^{ +1=} 2^{ +1=} 3^{ +1=} 4^{ +1=} 5^{ +1=} 6^{ +1=} 7^{ +1=} 8^{ +1=} 9^{ +1=} 10^{ +1=} 11^{ +1=} 12^{ +1=}...
1 + 1 = 2, 1 + 2 = 3, 2 + 3 = 5, 3 + 5 = 8, etc.
To put it this way:
1^{ +1=} 2^{ +1=} 3^{ +2=} 5^{ +3=} 8^{ +5=} 13^{ +8=} 21^{ +13=} 34^{ +21=} 55^{ +34=} 89^{ +55=} 144...
Furthermore, the ratios between the two Fibonacci numbers get closer to a certain ratio:
3 to 5 = 0.6
5 to 8 = 0.625
8 to 13 = 0.6153846
13 to 21 = 0.6190476
21 to 34 = 0.617647
34 to 55 = 0.6181818
55 to 89 = 0.6179775
89 to 144 = 0.6180555
144 to 233 = 0.6180257
233 to 377 = 0.6180371
377 to 610 = 0.6180327
610 to 987 = 0.6180344
987 to 1597 = 0.6180338
1597 to 2584 = 0.6180340
...
The number is equlivalent of (SQR(5)1)/2
The Fibonacci Golden Number is approximately 0.6180340
Amazing about the fact is that 1 + 0.6180340 = 1/0.6180340 = 1.6180340
Also, 0.6180340 + 0.6180340^{2} = 1
Natural Series: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,...
1^{ +2=}
3^{ +3=}
6^{ +4=}
10^{ +5=}
15^{ +6=}
21^{ +7=}
28^{ +8=}
36^{ +9=}
45^{ +10=}
55^{ +11=}
66^{ +12=}
78^{ +13=}
91^{ +14=}
105^{ +15=}
120...
Triangular Series: 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120,...
1^{ +3=} 4^{ +6=} 10^{ +10=} 20^{ +15=} 35^{ +21=} 56^{ +28=} 84^{ +36=} 120^{ +45=} 165^{ +55=} 220...
The Square Pyramid Series takes the list you created a bit further...
1^{ +4=} 5^{ +9=} 14^{ +16=} 30^{ +25=} 55^{ +36=} 91^{ +45=} 136...
just ones   1  

Natural   1  1  
Triangular   1  2  1  
Tetrahedal   1  3  3  1  
4D Tetrahedal   1  4  6  4  1  
5D Tetrahedal   1  5  10  10  5  1  
1  6  15  20  15  6  1  
1  7  21  35  35  21  7  1  
1  8  28  56  70  56  28  8  1  
1  9  36  84  126  126  84  36  9  1 
Odd: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21,...
1^{ +3=} 4^{ +5=} 9^{ +7=} 16^{ +9=} 25^{ +11=} 36^{ +13=} 49^{ +15=} 64^{ +17=} 81^{ +19=} 100^{ +21=} 121...
You may also notice that the Quad series are squares of the Natural numbers series:
1^{2}, 2^{2}, 3^{2}, 4^{2}, 5^{2}, 6^{2}, 7^{2}, 8^{2}, 9^{2}, 10^{2}, 11^{2}...
The sum of the first "n" odd numbers is a perfect square.
1+3+5+7+...+(2n1)=n^{2}
1, 5, 12, 22, 35, 51,...
1, 6, 18, 40, 75, 126,...
2, 6, 12, 20, 30, 42,...
Note that they're the doubling of the triangular series 1, 3, 6, 10,...
1^{3}+ 2^{3}= 9 = 3^{2}
1^{3}+ 2^{3}+ 3^{3}= 36 = 6^{2}
1^{3}+ 2^{3}+ 3^{3}+ 4^{3}= 100 = 10^{2}
1^{3}+ 2^{3}+ 3^{3}+ 4^{3}+ 5^{3}= 225 = 15^{2}
1^{3}+ 2^{3}+ 3^{3}+ ... n^{3} = (1+2+3+...n)^{2}
1+2+1=2^{2}
1+2+3+2+1=3^{2}
1  2  3  4  5  6 

2  6  12  20  30  
3  12  30  60  
4  20  60  
5  30  
6 

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