First you have the triangular numbers:

T(i) = 0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55

Then the square numbers both indexed with the first number i and j = 0

S(j) = 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100

It seems every natural number X:1...inf can be written as S(m)+T(n)+T(o) such that m+n+o is even for even X and odd for odd X...

for example:

15 = S(3) + T(2) +T(0)

next:

16 = S(4) + T(0) +T(0)

17 = S(4) + T(1) + T(0)

18 = S(4) +T(1) +T(1)

19 = S(3) + T(4) + T(0)

there's another way to write this but so m+n+o isn't odd

20 = S(3) + T(4) + T(1)

21 = S(3) + T(3) + T(3)

22 = S(0) + T(6) + T(0)

23 = S(4) + T(3) + T(0)

24 = S(3) + T(4) + T(3)

I don't have a proof but I think it's true for every natural number X....

## No comments:

## Post a Comment