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%I #10 Oct 31 2019 21:42:40
%S 1,4,2,17,9,3,72,38,5,12,305,161,8,51,6,1292,682,13,216,10,7,5473,
%T 2889,21,915,16,30,14,23184,12238,34,3876,26,127,59,25,98209,51841,55,
%U 16419,42,538,250,106,11,416020,219602,89,69552,68,2279,1059,449,18,33
%N Rectangular array R read by descending antidiagonals: divide to each even term of the Wythoff array (A035513) by 2, and delete all others.
%C Every positive integer occurs exactly once in R, and every row of R is a linear recurrence sequence. The appearance of a sequence s(r) below means that corresponding row of R is the same as s(r) except possibly for one or more initial terms of s(r).
%C Row 1 of R: A001076
%C Row 2 of R: A001077
%C Row 3 of R: A000045
%C Row 4 of R: A115179
%C Row 5 of R: A006355
%C Row 6 of R: A097924
%C Row 8 of R: A048875
%C Row 9 of R: A000032
%e Row 1 of the Wythoff array is (1,2,3,5,8,13,21,34,55,89,144,...), so that row 1 of R is (1,4,17,72,...).
%e _______________
%e Northwest corner of R:
%e 1 4 17 72 305 1292 5473
%e 2 9 38 161 682 2889 12238
%e 3 5 8 13 21 34 55
%e 12 51 216 915 3876 16419 69552
%e 6 10 16 26 42 68 110
%e 7 30 127 538 2279 9654 40895
%t w[n_, k_] := Fibonacci[k + 1] Floor[n*GoldenRatio] + (n - 1) Fibonacci[k];
%t Table[w[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten;
%t q[n_, k_] := If[Mod[w[n, k], 2] == 0, w[n, k]/2, 0];
%t t[n_] := Union[Table[q[n, k], {k, 1, 50}]];
%t u[n_] := If[First[t[n]] == 0, Rest[t[n]], t[n]]
%t Table[u[n], {n, 1, 10}] (* A328695 array *)
%t v[n_, k_] := u[n][[k]];
%t Table[v[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten (* A328695 sequence *)
%Y Cf. A035513, A001076, A001077, A000045, A115179, A006355, A097924, A048875, A000032, A328696, A328697.
%K nonn,tabl
%O 1,2
%A _Clark Kimberling_, Oct 26 2019
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