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,... | |

4-D 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 | ||||||||||||||||||||||||||||||||||

4-D Tetrahedal -- | 1 | 4 | 6 | 4 | 1 | |||||||||||||||||||||||||||||||||

5-D 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+...+(2n-1)=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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