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A328696
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.
3
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, 37, 16, 13, 117, 261, 134, 206, 96, 41, 21, 14, 189, 422, 350, 333, 155, 66, 54, 36, 25, 494, 1104, 566, 871, 405, 172, 87, 58, 40, 17, 799, 1786, 1481, 1409, 655
OFFSET
1,2
COMMENTS
Every positive integer occurs exactly once in R, and every row of R is a linear recurrence sequence.
EXAMPLE
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).
_______________
Northwest corner of R:
1, 2, 3, 7, 11, 28, 45, 117, 189, 494, 799
4, 6, 15, 24, 62, 100, 261, 422, 1104, 1786, 4675
5, 8, 20, 32, 83, 134, 350, 566, 1481, 2396, 6272
12, 19, 49, 79, 206, 333, 871, 1409, 3688, 5967, 15621
9, 23, 37, 96, 155, 405, 655, 1714, 2773, 7259, 11745
10, 16, 41, 66, 172, 278, 727, 1176, 3078, 4980, 13037
13, 21, 54, 87, 227, 367, 960, 1553, 4065, 6577, 17218
MATHEMATICA
w[n_, k_] := Fibonacci[k + 1] Floor[n*GoldenRatio] + (n - 1) Fibonacci[k];
Table[w[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten;
q[n_, k_] := If[Mod[w[n, k], 2] == 1, (1 + w[n, k])/2, 0];
t[n_] := Union[Table[q[n, k], {k, 1, 50}]];
u[n_] := If[First[t[n]] == 0, Rest[t[n]], t[n]]
s = Select[Range[40], ! u[#] == {} &]; u1[n_] := u[s[[n]]];
Column[Table[u1[n], {n, 1, 10}]] (* A328696 array *)
v[n_, k_] := u1[n][[k]];
Table[v[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten (* A328696 sequence *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Oct 26 2019
STATUS
approved