Pentagonal number

A pentagonal number is a Nombre illustrated which can be represented by a pentagon. A pentagonal number of row N is defined by

n (3n-1) \ over 2

N being a natural entirety not no one.

Examples

The first pentagonal numbers are

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

The pentagonal numbers are important in the theory of the divisions of entireties of Euler, and they intervene for example in its Théorème of the pentagonal number.

Generalization

The “generalized” pentagonal numbers are obtained starting from the formula given above, but with N taking values 0,1,-1,2,-2,3,-3,4…, producing the continuation

0,1,2,5,7,12,15,22,26,35,40,51,57,70,77,92,100,117,126,145,155,176,187,210,222,247,260,287,301,330,345,376,392,425,442,477,495,532,551,590,610,651,672,715,737,782,805,852,876,925,950,1001,1027

Remarks

The pentagonal number of row N is equal to a third of the triangular Nombre of row 3 N - 1.

The pentagonal numbers should not be confused with the centered pentagonal numbers.

See too

External bond

  • pentagonal Number

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