A182801 Joint-rank array of the numbers j*r^(i-1), where r = golden ratio = (1+sqrt(5))/2, i>=1, j>=1, read by antidiagonals.

1, 3, 2, 5, 6, 4, 7, 9, 11, 8, 10, 13, 16, 19, 14, 12, 18, 23, 28, 32, 25, 15, 21, 31, 39, 48, 54, 42, 17, 26, 36, 52, 66, 81, 89, 71, 20, 29, 44, 61, 86, 110, 134, 147, 117, 22, 34, 49, 73, 102, 141, 181, 221, 240, 193, 24, 38, 57, 82

Offset: 1

Author

Clark Kimberling, Dec 04 2010

Keywords

• nonn
• tabl

Comments

Joint-rank arrays are introduced here as follows.

Suppose that R={f(i,j)} is set of positive numbers, where i and j range through countable sets I and J, respectively, such that for every n, then number f(i,j) < n is finite. Let T(i,j) be the position of f(i,j) in the joint ranking of all the numbers in R. The joint-rank array of R is the array T whose i-th row is T(i,j).

For A182801, f(i,j)=j*r^(i-1), where r=(1+sqrt(5))/2 and I=J={1,2,3,...}.

(row 1)=A020959; (row 2)=A020960; (row 3)=A020961.

(col 1)=A020956; (col 2)=A020957; (col 3)=A020958.

Every positive integer occurs exactly once in A182801, so that as a sequence it is a permutation of the positive integers.

Examples

			Northwest corner:
1....3....5....7...10...12...
2....6....9...13...18...21...
4...11...16...23...31...36...
8...19...28...39...52...61...
		

Crossrefs

Cf. A001622, A182802, A182846, A182847, A182848, A182849, A252229.

Programs

Mathematica

r=GoldenRatio;
f[i_,j_]:=Sum[Floor[j*r^(i-k)],{k,1,i+Log[r,j]}];
TableForm[Table[f[i,j],{i,1,16},{j,1,16}]] (* A182801 *)

Formula

T(i,j)=Sum{floor(j*r^(i-k)): k>=1}.