login
Array read by descending antidiagonals: row k lists the numbers m such that 1/2^(k+1) < 1 - f(m) < 1/2^k, where f(m) is the fractional part of m*(golden ratio).
5

%I #16 Feb 27 2022 09:50:54

%S 2,4,1,5,6,3,7,9,11,16,10,14,24,29,8,12,17,32,50,42,21,13,19,37,63,97,

%T 76,55,15,22,45,71,131,110,199,288,18,27,53,84,152,165,343,521,144,20,

%U 30,58,105,186,254,432,665,754,377

%N Array read by descending antidiagonals: row k lists the numbers m such that 1/2^(k+1) < 1 - f(m) < 1/2^k, where f(m) is the fractional part of m*(golden ratio).

%C Every positive integer occurs exactly once, so that as a sequence, this is a permutation of the positive integers. The difference between consecutive terms in any row is a Fibonacci number, as is the difference between consecutive terms in column 1.

%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>

%e Upper-left corner:

%e 2 4 5 7 10 12 13 15 ...

%e 1 6 9 14 17 19 22 27 ...

%e 3 11 24 32 37 45 53 58 ...

%e 16 29 50 63 71 84 105 118 ...

%e 8 42 97 131 152 186 220 241 ...

%e 21 76 110 165 254 309 398 453 ...

%e ...

%t g = GoldenRatio; z = 50000; t = Table[N[FractionalPart[n*g]], {n, 1, z}];

%t r[k_] := Select[Range[z], (2^k - 1)/2^k < t[[#]] < (2*2^k - 1)/2^(k + 1) &];

%t s[n_] := Take[r[n], Min[20, Length[r[n]]]];

%t TableForm[Table[s[k], {k, 0, 10}]] (* A283741, array *)

%t w[i_, j_] := s[i][[j]]; Flatten[Table[w[n - k , k], {n, 10}, {k, n, 1, -1}]] (* A283741, sequence *)

%Y Cf. A000045, A001622, A283738, A283739, A283740.

%K nonn,tabl,easy

%O 1,1

%A _Clark Kimberling_, Mar 16 2017

%E Name corrected by _Jon E. Schoenfield_, Mar 25 2017