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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. 21
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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.

REFERENCES

Mohammad K. Azarian, Problem 123, Missouri Journal of Mathematical Sciences, Vol. 10, No. 3, Fall 1998, p. 176.  Solution published in Vol. 12, No. 1, Winter 2000, pp. 61-62.

LINKS

Table of n, a(n) for n=1..59.

FORMULA

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

EXAMPLE

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...

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 *)

CROSSREFS

Cf. A182802, A182846-A182849, A252229.

Sequence in context: A227192 A099889 A115511 * A026098 A135764 A253551

Adjacent sequences:  A182798 A182799 A182800 * A182802 A182803 A182804

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Dec 04 2010

STATUS

approved

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Last modified June 28 09:49 EDT 2017. Contains 288813 sequences.