Cantor's polynomials.
Bijections from Nⁿ to N.

The two only polynomials of degree 2, of the two variables x and y, which are bijections of N² on N, are the polynomials of Cantor (Cantor 1873. Shown in 1923 by Fueter and Pólya):

f(x,y)=((x+y)^2+3x+y)/2 et g(x,y)=((x+y)^2+x+3y)/2.

Naturally g(x,y)=f(y,x)

The general problem which is to determine the polynomials of degrees d bijective of Nⁿ on N remains open and seems very difficult.

By composing the preceding polynomials one obtains bijections of Nⁿ on N.

For example, the polynomials f(f(x, y), z), f(f(f(x, y), z), t) define bijections of N³ on N and N⁴ on N, but there are many others of them.

The table below gives some values of the polynomial: f(x,y)=((x+y)^2+3x+y)/2.

Starting from the couple (x, y) we can calculate his image k = f(x, y) in N.
Conversely, starting from k, we can calculate x and y tels que f(x, y) = k.

(See Éric Angelini's Decimation-like sequences).

Greedy algorithm for f_n(x, y):

Choose a step n>1.
In the half line N = {0, 1, 2, 3, 4, 5, 6 ...}, in free cases, from n places to n places (the step is n) and in the case K write f_n(0,y)= K (and K is no longer free). Then repeat the process for f_n(1,y), for f_n(,y) and so on.

n =     x y =        

This functions f_n permit to define integer Sequences :
Sequences f_n(x, 0) when x grows

f2(x, 0) : 0,1,3,7,15,31,63,127,255,511,1023,2047,4095,8191,16383,32767,... OEIS A000225

f3(x, 0) : 0,1,2,4,7,11,17,26,40,61,92,139,209,314,472,709,1064,1597,2396,3595,5393,8090,12136,18205,27308,40963,61445,... OEIS A006999

f4(x, 0) : 0,1,2,3,5,7,10,14,19,26,35,47,63,85,114,153,205,274,366,489,653,871,1162,1550,2067,2757,3677,4903,6538,8718,...

f5(x, 0) : 0,1,2,3,4,6,8,11,14,18,23,29,37,47,59,74,93,117,147,184,231,289,362,453,567,709,887,1109,1387,1734,2168,2711,3389,...

Sequences f_n(x, 1)

f2(x, 1) : 2,5,11,23,47,95,191,383,767,1535,3071,6143,12287,24575,49151,98303,...OEIS A055010 et A083329 (sauf le premier terme) 

f3(x, 1) : 3,5,8,13,20,31,47,71,107,161,242,364,547,821,1232,1849,2774,4162,6244,9367,14051,21077,31616,47425,71138,...

f4(x, 1) : 4,6,9,13,18,25,34,46,62,83,111,149,199,266,355,474,633,845,1127,1503,2005,2674,3566,4755,6341,8455,11274,...

Sequences f_n(x, x)

f2(x, x) : 0,5,19,55,143,351,831,1919,4351,9727,21503,47103,...

f3(x, x) : 0,5,16,34,67,121,223,382,655,1091,1840,2977,4858,7834,12775,20290,32663,51421,82484,...

f4(x, x) : 0,6,15,31,55,90,147,223,338,509,745,1077,1589,2274,3239,4671,6557,9255,13149,18467,...

Sequences f_n(1, y) y growing

f2(1, y) : 1,5,9,13,17,21,25,29,33,37,41,45,49,53,57,61,...OEIS A016813

f3(1, y) : 1,5,10,14,19,23,28,32,37,41,46,50,55,59,64,68,73,77,82,86,91,95,100,104,109,113,118,...

f4(1, y) : 1,6,11,17,22,27,33,38,43,49,54,59,65,70,75,81,86,91,97,102,107,113,118,123,129,134,139,145,150,155,161,166,171,177,182,187,193,...

Sequences f_n(2, y)

f2(2, y) : 3,11,19,27,35,43,51,59,67,75,83,91,99,107,115,123,...OEIS A017101 8y+3

f3(2, y) : 2,8,16,22,29,35,43,49,56,62,70,76,83,89,97,103,110,116,124,130,137,143,151,157,164,170,178,...

f4(2, y) : 2,9,15,23,30,37,45,51,58,66,73,79,87,94,101,109,115,122,130,137,143,151,158,165,173,179,186,194,201,207,215,222,229,237,243,250,258,...

See the link pages on Cantor and combinatorics.