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A335061
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Irregular table read by rows; n-th row corresponds to numbers in the range 0..2^n-1 whose binary expansion (possibly left-padded with 0's up to n binary digits) generates rotationally symmetric XOR-triangles.
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1
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0, 1, 0, 0, 2, 0, 6, 11, 13, 0, 14, 0, 30, 39, 57, 0, 8, 54, 62, 83, 91, 101, 109, 0, 126, 151, 233, 0, 40, 92, 116, 138, 162, 214, 254, 0, 72, 140, 196, 314, 370, 438, 510, 543, 599, 659, 731, 805, 877, 937, 993, 0, 168, 854, 1022, 1379, 1483, 1589, 1693
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refs;
listen;
history;
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internal format)
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OFFSET
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1,5
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COMMENTS
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LINKS
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Michael De Vlieger, Diagram montage showing 486 XOR-triangles T(n,k)>0 for 3 <= n <= 20.
Michael De Vlieger, Large 50X25 Diagram montage showing 1250 XOR-triangles T(n,k)>0 for 3 <= n <= 24.
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EXAMPLE
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The first rows are:
0, 1
0
0, 2
0, 6, 11, 13
0, 14
0, 30, 39, 57
0, 8, 54, 62, 83, 91, 101, 109
The XOR-triangles corresponding to the 8 terms of row 7 are (with dots instead of 0's for clarity):
T(7,1) = 0: T(7,2) = 8: T(7,3) = 54: T(7,4) = 62,
. . . . . . . . . . 1 . . . . 1 1 . 1 1 . . 1 1 1 1 1 .
. . . . . . . . 1 1 . . 1 . 1 1 . 1 1 . . . . 1
. . . . . . 1 . 1 . 1 1 . 1 1 1 . . . 1
. . . . 1 1 1 1 . 1 1 . 1 . . 1
. . . . . . 1 . 1 1 . 1
. . . . 1 1 1 1
. . . .
T(7,5) = 83: T(7,6) = 91: T(7,7) = 101: T(7,8) = 109:
1 . 1 . . 1 1 1 . 1 1 . 1 1 1 1 . . 1 . 1 1 1 . 1 1 . 1
1 1 1 . 1 . 1 1 . 1 1 . . 1 . 1 1 1 . 1 1 . 1 1
. . 1 1 1 . 1 1 . 1 1 1 1 . . 1 . 1 1 .
. 1 . . 1 . 1 1 . . 1 . 1 1 . 1
1 1 . 1 1 . . 1 1 . 1 1
. 1 . 1 1 . 1 .
1 1 1 1
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MATHEMATICA
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Table[Select[Range[0, 2^n - 1], Block[{k = #, w}, (Reverse /@ Transpose[#] /. -1 -> Nothing) == w &@ MapIndexed[PadRight[#, n, -1] &, Set[w, NestList[Map[BitXor @@ # &, Partition[#, 2, 1]] &, PadLeft[IntegerDigits[k, 2], n], n - 1]]]] &], {n, 12}] // Flatten (* Michael De Vlieger, May 24 2020 *)
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PROG
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(PARI) See Links section.
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CROSSREFS
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KEYWORD
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nonn,base,tabf
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AUTHOR
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STATUS
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approved
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