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%I #9 Oct 31 2019 21:12:02
%S 1,2,4,3,6,5,7,15,8,12,11,24,20,19,9,28,62,32,49,23,10,45,100,83,79,
%T 37,16,13,117,261,134,206,96,41,21,14,189,422,350,333,155,66,54,36,25,
%U 494,1104,566,871,405,172,87,58,40,17,799,1786,1481,1409,655
%N Rectangular array R read by descending antidiagonals: apply x -> (x+1)/2 to each odd term of the Wythoff array (A035513), and delete all others.
%C Every positive integer occurs exactly once in R, and every row of R is a linear recurrence sequence.
%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 (1,2,3,7,11,...) = A107857 (essentially).
%e _______________
%e Northwest corner of R:
%e 1, 2, 3, 7, 11, 28, 45, 117, 189, 494, 799
%e 4, 6, 15, 24, 62, 100, 261, 422, 1104, 1786, 4675
%e 5, 8, 20, 32, 83, 134, 350, 566, 1481, 2396, 6272
%e 12, 19, 49, 79, 206, 333, 871, 1409, 3688, 5967, 15621
%e 9, 23, 37, 96, 155, 405, 655, 1714, 2773, 7259, 11745
%e 10, 16, 41, 66, 172, 278, 727, 1176, 3078, 4980, 13037
%e 13, 21, 54, 87, 227, 367, 960, 1553, 4065, 6577, 17218
%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] == 1, (1 + 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 s = Select[Range[40], ! u[#] == {} &]; u1[n_] := u[s[[n]]];
%t Column[Table[u1[n], {n, 1, 10}]] (* A328696 array *)
%t v[n_, k_] := u1[n][[k]];
%t Table[v[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten (* A328696 sequence *)
%Y Cf. A035513, A107857, A328695, A328697.
%K nonn,tabl
%O 1,2
%A _Clark Kimberling_, Oct 26 2019
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