%I #21 Feb 22 2023 05:36:14

%S 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,

%T 54,42,17,26,36,52,66,81,89,71,20,29,44,61,86,110,134,147,117,22,34,

%U 49,73,102,141,181,221,240,193,24,38,57,82

%N 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.

%C Joint-rank arrays are introduced here as follows.

%C 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).

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

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

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

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

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

%e Northwest corner:

%e 1....3....5....7...10...12...

%e 2....6....9...13...18...21...

%e 4...11...16...23...31...36...

%e 8...19...28...39...52...61...

%t r=GoldenRatio;

%t f[i_,j_]:=Sum[Floor[j*r^(i-k)],{k,1,i+Log[r,j]}];

%t TableForm[Table[f[i,j],{i,1,16},{j,1,16}]] (* A182801 *)

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

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Dec 04 2010