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A340244
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Wythoff-B array read by antidiagonals.
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2
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4, 7, 12, 11, 20, 17, 18, 32, 28, 25, 29, 52, 45, 41, 33, 47, 84, 73, 66, 54, 38, 76, 136, 118, 107, 87, 62, 46, 123, 220, 191, 173, 141, 100, 75, 51, 199, 356, 309, 280, 228, 162, 121, 83, 59, 322, 576, 500, 453, 369, 262, 196, 134, 96, 67, 521, 932, 809
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OFFSET
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1,1
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COMMENTS
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The Wythoff array, A134859, consists of columns AA, BA, ABA, BBA, ABBA, BBBA, ... The Wythoff-B array consists of columns AAB, BAB, ABAB, BBAB, ABBAB, BBBAB, ... , formed by suffixing B to the column designations for A134859. Column k shows the numbers whose Zeckendorf representation has least terms F(k+1) and F(k+2), where F = A000045, the Fibonacci numbers. The rows are interspersed, in the sense that the order array (A340245) of the Wythoff-B array is an interspersion.
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LINKS
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FORMULA
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For n >=1 and k >= 1, w(n,k) = F(k-2)*A(A(B(n))) + F(k-1)*B(A(B(n))), where A(n) = floor(n*phi), B(n) = floor(n*phi^2); i.e., A = A000201, B= A001950, these being the lower and upper Wythoff sequences. (Note that F(-1) = 1, F(0) = 0.)
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EXAMPLE
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Corner:
4 7 11 18 29 47 76 123 199
12 20 32 52 84 136 220 356 576
17 28 45 73 118 191 309 500 809
25 41 66 107 173 280 453 733 1186
33 54 87 141 228 369 597 966 1563
38 62 100 162 262 424 686 1110 1796
46 75 121 196 317 513 830 1343 2173
51 83 134 217 351 568 919 1487 2406
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MATHEMATICA
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r = GoldenRatio; f[n_] := Fibonacci[n];
a[n_] := Floor[r*n]; b[n_] := Floor[r^2*n];
c[n_] := a[a[b[n]]]; d[n_] := b[a[b[n]]];
w[n_, k_] := f[k - 2] c[n] + f[k - 1] d[n];
Grid[Table[w[n, k], {n, 1, 15}, {k, 1, 15}]] (* A340244 array *)
Table[w[n - k + 1, k], {n, 15}, {k, n, 1, -1}] // Flatten (* A340244 sequence *)
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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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