cp's OEIS Frontend

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.

%I A182801 #21 Feb 22 2023 05:36:14
%S A182801 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 A182801 54,42,17,26,36,52,66,81,89,71,20,29,44,61,86,110,134,147,117,22,34,
%U A182801 49,73,102,141,181,221,240,193,24,38,57,82
%N 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.
%C A182801 Joint-rank arrays are introduced here as follows.
%C A182801 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 A182801 For A182801, f(i,j)=j*r^(i-1), where r=(1+sqrt(5))/2 and I=J={1,2,3,...}.
%C A182801 (row 1)=A020959; (row 2)=A020960; (row 3)=A020961.
%C A182801 (col 1)=A020956; (col 2)=A020957; (col 3)=A020958.
%C A182801 Every positive integer occurs exactly once in A182801, so that as a sequence it is a permutation of the positive integers.
%F A182801 T(i,j)=Sum{floor(j*r^(i-k)): k>=1}.
%e A182801 Northwest corner:
%e A182801 1....3....5....7...10...12...
%e A182801 2....6....9...13...18...21...
%e A182801 4...11...16...23...31...36...
%e A182801 8...19...28...39...52...61...
%t A182801 r=GoldenRatio;
%t A182801 f[i_,j_]:=Sum[Floor[j*r^(i-k)],{k,1,i+Log[r,j]}];
%t A182801 TableForm[Table[f[i,j],{i,1,16},{j,1,16}]] (* A182801 *)
%Y A182801 Cf. A001622, A182802, A182846, A182847, A182848, A182849, A252229.
%K A182801 nonn,tabl
%O A182801 1,2
%A A182801 _Clark Kimberling_, Dec 04 2010