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Natural interspersion of A000071(Fibonacci numbers minus 1), a rectangular array, by antidiagonals.
3

%I #5 Mar 30 2012 18:57:39

%S 1,2,3,4,5,6,7,8,9,10,12,13,14,15,11,20,21,22,23,16,17,33,34,35,36,24,

%T 25,18,54,55,56,57,37,38,26,19,88,89,90,91,58,59,39,27,28,143,144,145,

%U 146,92,93,60,40,41,29,232,233,234,235,147,148,94,61,62,42,30

%N Natural interspersion of A000071(Fibonacci numbers minus 1), a rectangular array, by antidiagonals.

%C 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, A194056 is a permutation of the positive integers; its inverse is A194057.

%e Northwest corner:

%e 1...2...4...7...12

%e 3...5...8...13..21

%e 6...9...14..22..35

%e 10..15..23..36..57

%e 11..16..24..37..58

%t z = 50;

%t c[k_] := -1 + Fibonacci[k + 2]

%t c = Table[c[k], {k, 1, z}] (* A000071, F(n+2)-1 *)

%t f[n_] := If[MemberQ[c, n], 1, 1 + f[n - 1]]

%t f = Table[f[n], {n, 1, 300}] (* A194055 *)

%t r[n_] := Flatten[Position[f, n]]

%t t[n_, k_] := r[n][[k]]

%t TableForm[Table[t[n, k], {n, 1, 7}, {k, 1, 7}]]

%t p = Flatten[Table[t[k, n - k + 1], {n, 1, 11}, {k, 1, n}]] (* A194056 *)

%t q[n_] := Position[p, n]; Flatten[Table[q[n], {n, 1, 70}]] (* A194057 *)

%Y Cf. A194029, A194055, A000071, A000045, A194057.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Aug 13 2011