A328695
Rectangular array R read by descending antidiagonals: divide to each even term of the Wythoff array (A035513) by 2, and delete all others.
Original entry on oeis.org
1, 4, 2, 17, 9, 3, 72, 38, 5, 12, 305, 161, 8, 51, 6, 1292, 682, 13, 216, 10, 7, 5473, 2889, 21, 915, 16, 30, 14, 23184, 12238, 34, 3876, 26, 127, 59, 25, 98209, 51841, 55, 16419, 42, 538, 250, 106, 11, 416020, 219602, 89, 69552, 68, 2279, 1059, 449, 18, 33
Offset: 1
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,...).
_______________
Northwest corner of R:
1 4 17 72 305 1292 5473
2 9 38 161 682 2889 12238
3 5 8 13 21 34 55
12 51 216 915 3876 16419 69552
6 10 16 26 42 68 110
7 30 127 538 2279 9654 40895
Cf.
A035513,
A001076,
A001077,
A000045,
A115179,
A006355,
A097924,
A048875,
A000032,
A328696,
A328697.
-
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] == 0, 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]]
Table[u[n], {n, 1, 10}] (* A328695 array *)
v[n_, k_] := u[n][[k]];
Table[v[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten (* A328695 sequence *)
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.
Original entry on oeis.org
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
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
-
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 *)
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