A138036 Write n = C(i,2)+C(j,1) with i>j>=0; let L[n] = [i,j]; sequence gives list of pairs L[n], n >= 0.

1, 0, 2, 0, 2, 1, 3, 0, 3, 1, 3, 2, 4, 0, 4, 1, 4, 2, 4, 3, 5, 0, 5, 1, 5, 2, 5, 3, 5, 4, 6, 0, 6, 1, 6, 2, 6, 3, 6, 4, 6, 5, 7, 0, 7, 1, 7, 2, 7, 3, 7, 4, 7, 5, 7, 6, 8, 0, 8, 1, 8, 2, 8, 3, 8, 4, 8, 5, 8, 6, 8, 7, 9, 0, 9, 1, 9, 2, 9, 3, 9, 4, 9, 5, 9, 6, 9, 7, 9, 8, 10, 0, 10, 1, 10, 2, 10, 3, 10, 4, 10, 5, 10, 6, 10, 7, 10, 8, 10, 9, 11, 0, 11, 1, 11, 2, 11, 3, 11, 4

Offset: 0

Author

N. J. A. Sloane, Sep 04 2011

Keywords

• nonn
• tabf

Comments

Each n >= 0 has a unique representation as n = C(i,2)+C(j,1) with i>j>=0. This is the combinatorial number system of degree t = 2. The i values are A002024, the j values A002262.

Examples

			The pairs L[0] through L[10] are
[1, 0]
[2, 0]
[2, 1]
[3, 0]
[3, 1]
[3, 2]
[4, 0]
[4, 1]
[4, 2]
[4, 3]
[5, 0]
		

References

• D. E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial Algorithms, Section 7.2.1.3, Eq. (20), p. 360.

Crossrefs

Cf. A002024, A002262. See A194847 for degree t=3.

Programs

Mathematica

A138036list[len_] := Module[{i = 0, j = 1, L = {1, 0}}, Do[i++; If[i == j, j++; i = 0]; AppendTo[L, j]; AppendTo[L, i], {len}]; L];
A138036list[60] (* Jean-François Alcover, Jul 11 2019, after Peter Luschny *)

Sage

def A138036_list(len):
    i, j = 0, 1
    L = [1, 0]
    for _ in range(len):
        i += 1
        if i == j:
            j += 1
            i = 0
        L.append(j)
        L.append(i)
    return L
A138036_list(47) # Peter Luschny, May 18 2015