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A191432
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Dispersion of ([nx+1/x]), where x=sqrt(2) and [ ]=floor, by antidiagonals.
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2
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1, 2, 5, 3, 7, 8, 4, 10, 12, 11, 6, 14, 17, 16, 15, 9, 20, 24, 23, 21, 18, 13, 28, 34, 33, 30, 26, 22, 19, 40, 48, 47, 43, 37, 31, 25, 27, 57, 68, 67, 61, 53, 44, 36, 29, 38, 81, 96, 95, 86, 75, 62, 51, 41, 32, 54, 115, 136, 135, 122, 106, 88, 72, 58, 45, 35, 77, 163, 193, 191, 173, 150, 125, 102, 82, 64, 50, 39
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table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,2
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COMMENTS
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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
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LINKS
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EXAMPLE
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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
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MATHEMATICA
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(* 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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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