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%I #9 Nov 02 2014 12:18:35
%S 1,2,3,4,6,8,5,7,9,11,10,12,14,16,21,13,15,17,19,24,29,18,20,22,27,32,
%T 37,42,23,25,30,35,40,45,50,55,26,28,33,38,43,48,53,58,63,31,36,41,46,
%U 51,56,61,66,71,76,34,39,44,49,54,59,64,69,74,79,84,47,52,57,62,67,72
%N Wythoff Triangle, T.
%C (1) Every positive integer occurs exactly once, so that
%C this is a permutation of the natural numbers.
%C (2) Obtained from the preliminary Wyhoff triangle
%C (A166309) by arranging each row in increasing order.
%C (3) The difference between consecutive row terms is a
%C Fibonacci number (A000045).
%C (4) Is the difference between consecutive column terms a
%C Fibonacci number?
%D C. Kimberling, "The Wythoff triangle and unique representations of positive integers," Proceedings of the Fourteenth International Conference on Fibonacci Numbers and Their Applications," Aportaciones Matematicas Invertigacion 20 (2011) 155-169.
%F For a=1,2,3,... and b=0,1,...,a-1, let P(a,b) be the
%F number of the row of the Wythoff array (A035513) that
%F precurses to (a,b). Then for each a, arrange the numbers P
%F (a,b) in increasing order.
%e The first nine rows of T:
%e 1
%e 2....3
%e 4....6...8
%e 5....7...9..11
%e 10..12..14..16..21
%e 13..15..17..19..24..29
%e 18..20..22..27..32..37..42
%e 23..25..30..35..40..45..50..55
%e 26..28..33..38..43..48..53..58..63
%e Row 5 of the preliminary Wythoff triangle is
%e 16,21,10,12,14, so that row 5 of the Wythoff triangle is
%e 10,12,14,16,21. These are the row numbers of the Wythoff
%e array W (A035513) which precurse to pairs (5,b) for
%e b=0,1,2,3,4, not respectively. Example of precursion: row
%e 16 of W is 40,65,105,...; then 65-40=25, 40-25=15,
%e 25-15=10, 15-10=5, 10-5=5, 5-5=0, 5-0=5, so that the
%e initial pair (5,0) is reached in seven precursive steps.
%t f[n_]:=f[n]=Fibonacci[n]; w[n_, k_] := f[k + 1] Floor[n GoldenRatio] + (n - 1) f[k]; a[n_, k_] := w[n, Module[{z = 0}, ((While[w[#1, z] <= w[#1, z + 1], z--]; z - 1) &)[n] + k]]; z = 100; t = Table[a[n, k], {n, 1, z}, {k, 1, 2}] (* n-th pair: 1st 2 terms of row n of left-justified Wythoff array, A165357 *)
%t u = Table[t[[n]][[1]], {n, 1, z}]
%t v = Table[Flatten[Position[u, n]], {n, 1, z/5}]
%t Flatten[v] (* A166310 sequence *)
%t TableForm[Table[Flatten[Position[u, n]], {n, 1, z/5}]] (* A166310 triangle, Clark Kimberling, Aug 01 2013 *)
%Y Cf. A035513, A165357, A166309.
%K nonn,tabl
%O 1,2
%A _Clark Kimberling_, Oct 11 2009
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