# Triangular and pyramidal numbers

Triangular and pyramidal numbers

Bill Gosper

f(n)=”The triangular numbers”=1+2+ . . .+n.Using 9 for n,

Graphics[{PointSize[Large],Point[Permutations[Range@9,{2}]]}]

I.e.,the unequal pairs of numbers up to 9 plots as a 9×9 square minus

its length 9 diagonal. But this is just two copies of 1+ . . .+8. So

its length 9 diagonal. But this is just two copies of 1+ . . .+8. So

2f(n-1)n-n;

2

Solving for f(n)

RSolve[%,f[n],n]//Factor

f[n]n(1+n)

1

2

This is essentially how the Gauß, the youngster, startled his grammar school teacher.

In highschool, my friend Reed Carson and I got this a different way—by adding up the areas of n+1 triangles:

In highschool, my friend Reed Carson and I got this a different way—by adding up the areas of n+1 triangles:

Table[{x,y},{x,4},{y,x}]

{{{1,1}},{{2,1},{2,2}},{{3,1},{3,2},{3,3}},{{4,1},{4,2},{4,3},{4,4}}}

Graphics[{EdgeForm[White],Blue,%/.{x_Integer,y_}->Rectangle[{x,y}]}]/.

{L___,Rectangle[{x_Integer,y_}]}{L,Triangle[{{x,y},{x+1,y},{x+1,y+1}}],

Red,Triangle[{{x,y},{x,y+1},{x+1,y+1}}]}

Now for 3D. Using as coordinates the integers 1, 2, ..., n taken three at a time,

Graphics3D[Sphere[Permutations[Range@9,{3}],1/2]]

The unequal triads of numbers up to 9 plots as a 9×9×9 cube minus

those diagonals which have two or three coordinates the same:

those diagonals which have two or three coordinates the same:

Graphics3D[Sphere[Select[Tuples[Range@9,3],Not[UnsameQ@@#]&],1/2]]

This is six triangulars of size 8 (= n-1) plus the central column of 9 (=n).

The former is six pyramids of size 7 (= n-2). Together they make a cube of size n=9.

So the equation for the pyramidal numbers solves as

The former is six pyramids of size 7 (= n-2). Together they make a cube of size n=9.

So the equation for the pyramidal numbers solves as

RSolve[6f[n-2]+6n(n-1)/2+nn^3,f[n],n]//Factor

f[n]n(1+n)(2+n)

1

6

Instead of 1 + 3 + 6 + . . . + n, summing 1² + 2² + . . . + n² gives the “square pyramidal numbers”,

Table[Sphere[{x,y,z},1/2],{z,7},{y,z},{x,z}]//Graphics3D

each of which is the sum of two consecutive ordinary pyramidal numbers:

%/.Sphere[{x_,y_,z_},1/2]Sphere[{x,y,z},1/4]/;x<y

So the sum of the first n squares, 1² + 2² + . . . + n², is