A020957 -id:A020957 - 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.

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

Clark Kimberling, Dec 04 2010

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.

			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...

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

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 *)

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

A020958 a(n) = Sum_{k >= 1} floor(3*tau^(n-k)).

Original entry on oeis.org

5, 9, 16, 28, 48, 81, 134, 221, 361, 589, 957, 1554, 2519, 4082, 6610, 10702, 17322, 28035, 45368, 73415, 118795, 192223, 311031, 503268, 814313, 1317596, 2131924, 3449536, 5581476, 9031029, 14612522, 23643569, 38256109, 61899697, 100155825, 162055542, 262211387

Offset: 1

Clark Kimberling

nonn

Since 3*tau^(-3) < 1 the number of nonzero terms in the sum is finite. - Giovanni Resta, Jul 08 2019

C. Kimberling, Problem 10520, Amer. Math. Mon. 103 (1996) p. 347.

Cf. A001622 (tau), A020957.

a[n_] := Sum[Floor[3 GoldenRatio^k], {k, -2, n-1}]; Array[a, 37] (* Giovanni Resta, Jul 08 2019 *)

a(n) = Sum_{k=-2..(n-1)} floor(3*tau^k). - Giovanni Resta, Jul 08 2019

Name edited by Michel Marcus, Jul 06 2019

a(27)-a(37) from Giovanni Resta, Jul 08 2019

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.

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