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Search: id:A120710
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| A120710 |
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A GF(2) polynomial analog of triangular numbers. |
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1
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| 0, 0, 0, 2, 0, 4, 8, 14, 0, 8, 16, 26, 32, 44, 56, 70, 0, 16, 32, 50, 64, 84, 104, 126, 128, 152, 176, 202, 224, 252, 280, 310, 0, 32, 64, 98, 128, 164, 200, 238, 256, 296, 336, 378, 416, 460, 504, 550, 512, 560, 608, 658, 704, 756, 808, 862, 896, 952, 1008, 1066, 1120
(list; graph; listen)
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OFFSET
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0,4
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COMMENT
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The k-th bit in a(n) is one just if there are an odd number of pairs of distinct one bits i#j in n such that i+j=k. GF(2) polynomial ("XOR numbral") multiplication can be implemented as A048720(i,j) = A000695(i AND j) XOR a(i AND j) XOR a(i IOR j) XOR a(i AND NOT j) XOR a(NOT i AND j), analogously to ordinary multiplication (A003991) ij = tri(i+j)-tri(i)-tri(j) via triangular numbers (A000217).
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REFERENCES
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Posting by Richard Schroeppel (rschroe(AT)sandia.gov) to math-fun mailing list, Jun 26 2006.
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FORMULA
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a(0)=0; a(n + 2^k) = a(n) XOR (n * 2^k), 0<=n<2^k.
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EXAMPLE
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a(15)=54 because 15=2^0+2^1+2^2+2^3, the four one-bits giving six distinct pairs 01 02 03 12 13 23, which sum to 1 2 3 3 4 5, of which 1 2 4 and 5 occur oddly, yielding 2^1+2^2+2^4+2^5=54.
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CROSSREFS
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Cf. A048720, A000695, A003991, A000217.
Sequence in context: A021087 A120558 A120554 this_sequence A115780 A101189 A070015
Adjacent sequences: A120707 A120708 A120709 this_sequence A120711 A120712 A120713
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KEYWORD
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base,easy,nonn
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AUTHOR
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Marc LeBrun (mlb(AT)well.com), Jun 28 2006
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