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A137707 Secondary Wythoff Array read by descending antidiagonals.

%I A137707 #8 May 14 2025 00:26:39
%S A137707 1,3,2,5,4,7,9,6,13,8,15,10,21,14,11,25,16,35,22,19,12,41,26,57,36,31,
%T A137707 20,17,67,42,93,58,51,32,29,18,109,68,151,94,83,52,47,30,23,177,110,
%U A137707 245,152,135,84,77,48,39,24,286,178,397,246,220,136,125,78,63,40,27
%N A137707 Secondary Wythoff Array read by descending antidiagonals.
%C A137707 (1) Delete the odd numbered rows and get twice the Wythoff array, A035513.
%C A137707 (2) Subtract 1 from the even numbered rows and get the odd numbered rows.
%C A137707 (3) As a sequence, this is a permutation of the positive integers.
%C A137707 (4) The array is a dispersion and an interspersion.
%C A137707 (5) Let c = ordered union of odd numbered columns and let d = ordered union of even numbered columns; then c and d are the unique solutions of the complementary equation d(n)=c(c(n))+2 and also of the complementary equation d(n)=c(n)+2*Floor[(n+2)/2].
%C A137707 (6) c=A137708, d=A137709.
%F A137707 Odd numbered rows: r(n)=r(n-1)+r(n-2)+1, Even numbered rows: r(n)=r(n-1)+r(n-2).
%e A137707 Corner:
%e A137707     1    3    5     9    15    25    41    67    109   177
%e A137707     2    4    6    10    16    26    42    68    110   178
%e A137707     7   13   21    35    57    93   151   245    397   643
%e A137707     8   14   22    36    58    94   152   246    398   644
%e A137707    11   19   31    51    83   135   219   355    575   931
%e A137707    12   20   32    52    84   136   220   356    576   932
%e A137707    17   29   47    77   125   203   329   533    863  1397
%e A137707    18   30   48    78   126   204   330   534    864  1398
%e A137707    23   39   63   103   167   271   439   711   1151  1863
%e A137707    24   40   64   104   168   272   440   712   1152  1864
%e A137707    27   45   73   119   193   313   507   821   1329  2151
%e A137707    28   46   74   120   194   314   508   822   1330  2152
%e A137707    33   55   89   145   235   381   617   999   1617  2617
%t A137707 w[n_, k_] := Fibonacci[k + 1]  Floor[n*GoldenRatio] + (n - 1)  Fibonacci[k];
%t A137707 t = Table[2 w[n, k], {n, 1, 12}, {k, 1, 12}]; Grid[Riffle[t - 1, t]]
%t A137707  (* _Clark Kimberling_, May 09 2025 *)
%Y A137707 Cf. A035513, A137708, A137709.
%K A137707 nonn,tabl
%O A137707 1,2
%A A137707 _Clark Kimberling_, Feb 07 2008