cp's OEIS Frontend

A128440 Array T(n,k) = floor(k*t^n) where t = golden ratio = (1 + sqrt(5))/2, read by descending antidiagonals.

%I A128440 #31 Nov 14 2022 09:01:55
%S A128440 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 A128440 29,12,18,25,34,44,53,58,46,14,20,29,41,55,71,87,93,76,16,23,33,47,66,
%U A128440 89,116,140,152,122,17,26,38,54,77,107,145,187,228,245,199
%N A128440 Array T(n,k) = floor(k*t^n) where t = golden ratio = (1 + sqrt(5))/2, read by descending antidiagonals.
%C A128440 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 A128440 Conjecture:  rows 2n-1 and 2n are disjoint for every positive integer n. - _Clark Kimberling_, Nov 11 2022
%C A128440 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 A128440 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 A128440 T(k,n) = k*F(n-1) + floor(k*t*F(n)), where F=A000045, the Fibonacci numbers.
%e A128440 Corner:
%e A128440    1    3    4    6    8    9   11   12
%e A128440    2    5    7   10   13   15   18   20
%e A128440    4    8   12   16   21   25   29   33
%e A128440    6   13   20   27   34   41   47   54
%e A128440   11   22   33   44   55   66   77   88
%e A128440   17   35   53   71   89  107  125  143
%e A128440   29   58   87  116  145  174  203  232
%e A128440   46   93  140  187  234  281  328  375
%t A128440 r = (1 + Sqrt[5])/2; t[k_, n_] := Floor[n*r^k];
%t A128440 Grid[Table[t[k, n], {k, 1, 10}, {n, 1, 20}]]
%t A128440 (* _Clark Kimberling_, Nov 11 2022 *)
%o A128440 (PARI) T(n,k) = floor(k*quadgen(5)^n);
%o A128440 matrix(7, 7, n, k, T(n,k)) \\ _Michel Marcus_, Nov 14 2022
%Y A128440 Cf. A000045, A001622, A000201, A001950, A128439, A358359.
%K A128440 nonn,tabl
%O A128440 1,2
%A A128440 _Clark Kimberling_, Mar 03 2007