OFFSET

1,2

COMMENTS

Background discussion: Suppose that s is an increasing sequence of positive integers, that the complement t of s is infinite, and that t(1)=1. The dispersion of s is the array D whose n-th row is (t(n), s(t(n)), s(s(t(n)), s(s(s(t(n)))), ...). Every positive integer occurs exactly once in D, so that, as a sequence, D is a permutation of the positive integers. The sequence u given by u(n)=(number of the row of D that contains n) is a fractal sequence.

Examples:

Conjecture: It appears this sequence is related to the even numbers with odd abundance A088827. Looking at the table format if the columns represent the powers of 2 (starting at 2^1) and the rows represent the squares of odd numbers, then taking the product of a term's row and column gives the n-th term in A088827. Example: A088827(67) = (7^2) * (2^6) = 3136. - John Tyler Rascoe, Jul 12 2022

LINKS

Michel Marcus, Antidiagonals n = 1..100, flattened

EXAMPLE

Northwest corner:

1 2 3 4 6 9

5 7 10 14 20 28

8 12 17 24 34 48

11 16 23 33 47 67

15 21 30 43 61 86

MATHEMATICA

(* Program generates the dispersion array T of increasing sequence f[n] *)

r = 40; r1 = 12; (* r=# rows of T, r1=# rows to show *)

c = 40; c1 = 12; (* c=# cols of T, c1=# cols to show *)

x = Sqrt[2];

f[n_] := Floor[n*x + 1/x] (* f(n) is complement of column 1 *)

mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1,

Length[Union[list]]]

rows = {NestList[f, 1, c]};

Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];

t[i_, j_] := rows[[i, j]]; TableForm[

Table[t[i, j], {i, 1, 10}, {j, 1, 10}]] (* A191432 array *)

Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191432 sequence *)

(* Program by Peter J. C. Moses, Jun 01 2011 *)

PROG

(PARI) s(n) = my(x=quadgen(8)); floor(n*x+1/x); \\ A001953

t(n) = floor((n+1/2)*(2+quadgen(8))); \\ A001954

T(n, k) = my(x = t(n-1)); for (i=2, k, x = s(x); ); x; \\ Michel Marcus, Jul 13 2022

CROSSREFS

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Jun 03 2011

STATUS

approved