some questions on mapping from an integer to a vector and inversely

[American Citizen <website.reader3@gmail.com>]

Hello:

My topic is on how to develop mapping from one integer n into n-integers 
[vector] , or 1-dimensional space to n-dimensional space.

This is a mapping of 1d(n) --> nd[n] and of course, the inverse map 
which takes the n-dim vector back to the integer.

When writing C or C++ code I commonly use loops for variables such as

for ( a = 0; a<100; a++) {

  for( b = a+1; b<100; b++) {

    do stuff

  }

}

We want a unique n for {a,b}

Other times I start the loop at 1 not 0 as the root index

for ( i = 1, n, for(j=i+1,n, {do stuff} ) )

So similarly we want a unique n. I call these loops zero-based or 
one-based, which is very similar to how C or C++ code indexes vectors 
(which is zero-based)

I developed some simple maps on my own

map_xy_to_n(x,y) = (y^2 + (2*x + 1)*y + x*(x + 3))/2;
map_n_to_xy(n) = 
S=floor((sqrt(8*n+1)-1)/2);T=S*(S+1)/2;x=n-T;y=S-x;return([x,y]);
bi_Z_to_N(Z)=if(Z<0,return(-2*Z-1),return(2*Z));
bi_N_to_Z(N)=if(N%2==0,return(N/2),return(-(N+1)/2));

\\ Forward map x,y --> n

\\ 21 [0,6]
\\ 15 [0,5] 22 [1,5]
\\ 10 [0,4] 16 [1,4] 23 [2,4]
\\  6 [0,3] 11 [1,3] 17 [2,3] 24 [3,3]
\\  3 [0,2]  7 [1,2] 12 [2,2] 18 [3,2] 25 [4,2]
\\  1 [0,1]  4 [1,1]  8 [2,1] 13 [3,1] 19 [4,1] 26 [5,1]
\\  0 [0,0]  2 [1,0]  5 [2,0]  9 [3,0] 14 [4,0] 20 [5,0] 27 [6,0]

\\ Inverse map n --> x,y

\\ 21 [0,6] 22 [1,5] 23 [2,4] 24 [3,3] 25 [4,2] 26 [5,1] 27 [6,0]
\\ 15 [0,5] 16 [1,4] 17 [2,3] 18 [3,2] 19 [4,1] 20 [5,0]
\\ 10 [0,4] 11 [1,3] 12 [2,2] 13 [3,1] 14 [4,0]
\\  6 [0,3]  7 [1,2]  8 [2,1]  9 [3,0]
\\  3 [0,2]  4 [1,1]  5 [2,0]
\\  1 [0,1]  2 [1,0]
\\  0 [0,0]

xy_MAP0_n(V)=my(x,y);[x,y]=V;x*(x-1)/2+y;
n_MAP0_xy(n)=x=floor(real(polroots('x*('x-1)/2-n)[2]));y=n-x*(x-1)/2;[x,y];
xy_MAP1_n(V)=xy_MAP0_n([V[1]-1,V[2]-1])+1;
n_MAP1_xy(n)=[x,y]=n_MAP0_xy(n-1);[x+1,y+1];

but I want to move to 3 or 4 terms involved, not just 2.

Is there an easy way to do 1-d mapping to n-d and the inverse, of course?

Randall