Pentagonal Numbers

Geometric Numbers

look at the pentagonal sequence, the difference between numbers, and then the differences between differences
1,5,12,22,35,51,70,..... becomes
1,4,7,10,13,16,19,.......this becomes
3,3,3,3,3......
so each number grows by a 4 and some multiple of 3.

returning to the sequence
1 P(1)=1
2 P(2)=5=1+4
3 P(3)=12=5+4+3
4 P(4)=22=12+4+3+3
5 P(5)=35=22+4+3+3+3
6 P(6)=51=22+4+(6-2)*3
7 P(7)=70=51+4+(7-2)*3
.
.
n P(n)=P(n-1)+4+(n-2)*3

and the difference between P(n) and P(n-1)?
[P(n)-P(n-1)]=4+(n-2)*3=4+3n-8=3n-5

So
P(n+1)=P(n)+3(n+1)-5

like triangular numbers
T(n)=n(n-1)/2

pentagonal numbers also end up in a series
P(n)=n(3n-1)/2

Rap

I don't know how I thinking this morning

P(n)=P(n-1)+4+(n-2)*3 =P(n-1)+4+3n-6=p(n-1)+3n-2
so
P(n)-P(n-1)=3n-2

Rap

Thanks!