cp's OEIS Frontend

A194030 Natural interspersion of the Fibonacci sequence (1,2,3,5,8,...), a rectangular array, by antidiagonals.

%I A194030 #34 May 18 2025 07:57:16
%S A194030 1,2,4,3,6,7,5,9,10,11,8,14,15,16,12,13,22,23,24,17,18,21,35,36,37,25,
%T A194030 26,19,34,56,57,58,38,39,27,20,55,90,91,92,59,60,40,28,29,89,145,146,
%U A194030 147,93,94,61,41,42,30,144,234,235,236,148,149,95,62,63,43,31
%N A194030 Natural interspersion of the Fibonacci sequence (1,2,3,5,8,...), a rectangular array, by antidiagonals.
%C A194030 See A194029 for definitions of natural fractal sequence and natural interspersion. Every positive integer occurs exactly once (and every pair of rows intersperse), so that as a sequence, this sequence is a permutation of the positive integers; its inverse is A194031.
%H A194030 Michael De Vlieger, <a href="/A194030/b194030.txt">Table of n, a(n) for n = 1..11325</a> (rows n = 1..150, flattened).
%H A194030 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A194030 T(n,k) = Fibonacci(A130233(n) + k - 1) + n - 1. - _Michael De Vlieger_, May 17 2025
%e A194030 Northwest corner:
%e A194030   1...2...3...5...8...13
%e A194030   4...6...9...14..22..35
%e A194030   7...10..15..23..36..57
%e A194030   11..16..24..37..58..92
%e A194030   12..17..25..38..59..93
%t A194030 z = 40;
%t A194030 c[k_] := Fibonacci[k + 1];
%t A194030 c = Table[c[k], {k, 1, z}]  (* A000045 *)
%t A194030 f[n_] := If[MemberQ[c, n], 1, 1 + f[n - 1]]
%t A194030 f = Table[f[n], {n, 1, 800}]  (* A194029 *)
%t A194030 r[n_] := Flatten[Position[f, n]]
%t A194030 t[n_, k_] := r[n][[k]]
%t A194030 TableForm[Table[t[n, k], {n, 1, 8}, {k, 1, 7}]]
%t A194030 p = Flatten[Table[t[k, n - k + 1], {n, 1, 13}, {k, 1, n}]]  (* A194030 *)
%t A194030 q[n_] := Position[p, n]; Flatten[Table[q[n], {n, 1, 80}]]  (* A194031 *)
%t A194030 (* Second program: *)
%t A194030 f[x_] := Block[{i = 1}, While[x > Fibonacci[i], i++]; i - 1]; Table[Fibonacci[f[k] + n - k + 1] + k - 1, {n, 2, 12}, {k, n - 1}] // Flatten (* _Michael De Vlieger_, May 17 2025 *)
%Y A194030 Cf. A130233, A194029, A194031 (inverse).
%Y A194030 Column 1 appears to be A345347.
%Y A194030 Although A383977 initially has many terms in comment with this, the author of A383977 has observed that the two sequences are unrelated. - _N. J. A. Sloane_, May 17 2025
%K A194030 nonn,tabl
%O A194030 1,2
%A A194030 _Clark Kimberling_, Aug 12 2011