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A361994
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(2,2)-block array, B(2,1), of the Wythoff array (A035513), read by descending antidiagonals.
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4
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14, 37, 40, 97, 105, 69, 254, 275, 181, 95, 665, 720, 474, 249, 124, 1741, 1885, 1241, 652, 325, 150, 4558, 4935, 3249, 1707, 851, 393, 179, 11933, 12920, 8506, 4469, 2228, 1029, 469, 205, 31241, 33825, 22269, 11700, 5833, 2694, 1228, 537, 234, 81790, 88555
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OFFSET
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1,1
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COMMENTS
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We begin with a definition. Suppose that W = (w(i,j)), where i >= 1 and j >= 1, is an array of numbers such that if m and n satisfy 1 <= m < n, then there exists k such that w(m,k+h) < w(n,h+1) < w(m,k+h+1) for every h >= 0. Then W is a row-splitting array. The array B(2,2) is a row-splitting array. The rows of B(2,2) are linearly recurrent with signature (3,-1); the columns are linearly recurrent with signature (1,1,-1). The order array (as defined in A333029) of B(2,2) is A361996.
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LINKS
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FORMULA
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B(2,2) = (b(i,j)), where b(i,j) = w(2i-1,2j-1) + w(2i-1,2j) + w(2i,2j-1) + w(2i,2j) for i >= 1, j >= 1, where (w(i,j)) is the Wythoff array (A035513).
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EXAMPLE
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Corner of B(2,2):
14 37 97 254 665 1741 ...
40 105 275 720 1885 4935 ...
69 181 474 1241 3249 8506 ...
95 249 652 1707 4469 11700 ...
124 325 851 2228 5833 15271 ...
...
b(1,1) = w(1,1) + w(1,2) + w(2,1) + w(2,2) = 1 + 2 + 4 + 7 = 14;
b(1,2) = w(1,3) + w(1,4) + w(2,3) + w(2,4) = 3 + 5 + 11 + 18 = 37;
b(2,1) = w(3,1) + w(3,2) + w(4,1) + w(4,2) = 8 + 10 + 9 + 15 = 40.
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MATHEMATICA
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f[n_] := Fibonacci[n]; r = GoldenRatio;
zz = 10; z = 13;
w[n_, k_] := f[k + 1] Floor[n*r] + (n - 1) f[k]
t[h_, k_] := w[2 h - 1, 2 k - 1] + w[2 h - 1, 2 k] + w[2 h, 2 k - 1] + w[2 h, 2 k];
Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten (*A361994 sequence *)
TableForm[Table[t[h, k], {h, 1, zz}, {k, 1, z}]] (* A361994 array *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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