A020959 -id:A020959 - 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}.

A347065 Rectangular array (T(n,k)), by antidiagonals: T(n,k) = position of k in the ordering of {h/r^m, r = (1+sqrt(5))/2, h >= 1, 0 <= m <= n}.

Original entry on oeis.org

1, 3, 1, 4, 3, 1, 6, 5, 3, 1, 8, 7, 5, 3, 1, 9, 9, 7, 5, 3, 1, 11, 11, 10, 7, 5, 3, 1, 12, 13, 12, 10, 7, 5, 3, 1, 14, 15, 14, 12, 10, 7, 5, 3, 1, 16, 17, 16, 15, 12, 10, 7, 5, 3, 1, 17, 19, 19, 17, 15, 12, 10, 7, 5, 3, 1, 19, 21, 21, 20, 17, 15, 12, 10, 7

Offset: 1

Clark Kimberling, Aug 16 2021

			Corner:
   1 3 4 6  8  9 11 12 14 16 17 19 21
   1 3 5 7  9 11 13 15 17 19 21 23 25
   1 3 5 7 10 12 14 16 19 21 23 25 28
   1 3 5 7 10 12 15 17 20 22 24 26 29
   1 3 5 7 10 12 15 17 20 22 24 27 30
   1 3 5 7 10 12 15 17 20 22 24 27 30
   1 3 5 7 10 12 15 17 20 22 24 27 30

Cf. A000201 (row 1), A005408 (row 2), A190511 (row 3), A020959 (limiting row).

Cf. A347066, A347067, A347068, A347069.

z = 100; r = N[(1 + Sqrt[5])/2];
s[m_] := Range[z] r^m; t[0] = s[0];
t[n_] := Sort[Union[s[n], t[n - 1]]]
row[n_] := Flatten[Table[Position[t[n], N[k]], {k, 1, z}]]
TableForm[Table[row[n], {n, 1, 10}]] (* A347065, array *)
w[n_, k_] := row[n][[k]];
Table[w[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten (* A347065, sequence *)

A252229 The number of numbers j*r^k in the interval [n,n+1), where r = (1 + sqrt(5))/2, the golden ratio, and j >=0, k >= 0.

Original entry on oeis.org

1, 2, 2, 2, 3, 2, 3, 2, 3, 2, 2, 3, 3, 3, 2, 2, 3, 3, 2, 2, 3, 3, 3, 2, 2, 3, 2, 3, 2, 4, 2, 2, 2, 4, 3, 3, 2, 2, 3, 2, 2, 3, 3, 2, 3, 2, 4, 3, 2, 2, 3, 2, 2, 3, 3, 4, 2, 2, 3, 3, 2, 3, 2, 3, 2, 2, 3, 3, 3, 2, 2, 3, 3, 2, 2, 3, 4, 3, 2, 2, 3, 2, 3, 2, 3, 2

Offset: 0

Clark Kimberling, Dec 16 2014

The least n for which a(n) = 4 is 29; the least n for which a(n) = 5 is 199.

			in [0,1):  0
in [1,2):  1, 1 + r
in [2,3):  2, 2 + r
in [3,4):  3, 1+2*r
in [4,5):  4, 1+3*r, 2 + r

Clark Kimberling, Table of n, a(n) for n = 0..1000

Cf. A182801, A020959.

z = 100; r = (1 + Sqrt[5])/2;
s[n_, j_] := s[n, j] = Floor[Log[n/j]/Log[r]];
a[n_] := a[n] = Sum[s[n + 1, j] - s[n, j], {j, 1, Floor[(n + 1)/r]}];
t = Join[{1}, Table[1 + a[n], {n, 1, z}]] (* A252229 *)

a(n) = 1 + sum{s(n+1,j) - s(n,j), j=1..floor[(n+1)/r]}, where s(n,j) = floor[log(n/j)/log(r)], for n >= 1.

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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A347065 Rectangular array (T(n,k)), by antidiagonals: T(n,k) = position of k in the ordering of {h/r^m, r = (1+sqrt(5))/2, h >= 1, 0 <= m <= n}.

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A252229 The number of numbers j*r^k in the interval [n,n+1), where r = (1 + sqrt(5))/2, the golden ratio, and j >=0, k >= 0.

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