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%I #31 Nov 14 2022 09:01:55
%S 1,3,2,4,5,4,6,7,8,6,8,10,12,13,11,9,13,16,20,22,17,11,15,21,27,33,35,
%T 29,12,18,25,34,44,53,58,46,14,20,29,41,55,71,87,93,76,16,23,33,47,66,
%U 89,116,140,152,122,17,26,38,54,77,107,145,187,228,245,199
%N Array T(n,k) = floor(k*t^n) where t = golden ratio = (1 + sqrt(5))/2, read by descending antidiagonals.
%C Row 1 = Lower Wythoff sequence = A000201; Row 2 = Upper Wythoff sequence = A001950; Column 1 = A014217 (after first term); T(n,n) = A128440(n). Every positive integer occurs exactly once in the first two rows.
%C Conjecture: rows 2n-1 and 2n are disjoint for every positive integer n. - _Clark Kimberling_, Nov 11 2022
%C Stronger conjecture: for any positive integer n, if the numbers in rows 2n-1 and 2n are jointly arranged in increasing order, and each number is replaced by its position in the ordering, then the resulting two rows are identical to the first two rows. - _Clark Kimberling_, Nov 13 2022
%H Michel Marcus, <a href="/A128440/b128440.txt">Table of n, a(n) for n = 1..5050</a> (Antidiagonals n=1..100 of array, flattened).
%F T(k,n) = k*F(n-1) + floor(k*t*F(n)), where F=A000045, the Fibonacci numbers.
%e Corner:
%e 1 3 4 6 8 9 11 12
%e 2 5 7 10 13 15 18 20
%e 4 8 12 16 21 25 29 33
%e 6 13 20 27 34 41 47 54
%e 11 22 33 44 55 66 77 88
%e 17 35 53 71 89 107 125 143
%e 29 58 87 116 145 174 203 232
%e 46 93 140 187 234 281 328 375
%t r = (1 + Sqrt[5])/2; t[k_, n_] := Floor[n*r^k];
%t Grid[Table[t[k, n], {k, 1, 10}, {n, 1, 20}]]
%t (* _Clark Kimberling_, Nov 11 2022 *)
%o (PARI) T(n,k) = floor(k*quadgen(5)^n);
%o matrix(7, 7, n, k, T(n,k)) \\ _Michel Marcus_, Nov 14 2022
%Y Cf. A000045, A001622, A000201, A001950, A128439, A358359.
%K nonn,tabl
%O 1,2
%A _Clark Kimberling_, Mar 03 2007
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