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A201982
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Consider the numbers 0 <= n <= 999 whose decimal digits are represented by (a,b,c). Look at the cross product of the vectors (u,v,w) = (a,b,c)^(c,b,a) in 3-dimensional Euclidean space. Sequence gives numbers n such that the components u, v, w are > = 0.
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1
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0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 30, 40, 50, 60, 70, 80, 90, 101, 102, 103, 104, 105, 106, 107, 108, 109, 111, 121, 131, 141, 151, 161, 171, 181, 191, 202, 203, 204, 205, 206, 207, 208, 209, 212, 222, 232, 242, 252, 262, 272, 282, 292, 303, 304, 305, 306
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OFFSET
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1,3
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COMMENTS
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The sequence contains 145 numbers. The numbers < 100 are represented by the form (0,0,0) or (0,0,x) or (0,x,y). The subset of palindromes with three decimal digits of A002113 is included in this sequence.
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LINKS
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FORMULA
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From the classical formula of the cross product (or vector product) of two vectors: U = (u1,u2,u3) and V = (v1,v2,v3) with U^V = (u2*v3 - u3*v2, u3*v1 - u1*v3, u1*v2 - u2*v1), we obtain (a,b,c)^(c,b,a) = (a*b-b*c, c^2-a^2, a*b-b*c).
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EXAMPLE
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2 is in the sequence because (0,0,2)^(2,0,0) = (0*0-2*0, 2*2-0*0, 0*0-0*2) = (0,4,0);
509 is in the sequence because (5,0,9)^(9,0,5) = (0*5-0*9, 9*9-5*5, 5*0-0*9) = (0,56,0).
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MAPLE
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V:=array(1..3):for n from 0 to 999 do: V[1]:=0: V[2]:=0: V[3]:=0:W:=convert(n, base, 10): if nops(W)=1 then V[1]:=W[1]:else fi:if nops(W)=2 then V[1]:=W[1]: V[2]:=W[2]:else fi:if nops(W)=3 then V[1]:=W[1]: V[2]:=W[2]: V[3]:=W[3]:else fi: if V[3] * V[2] - V[2] * V[1] >= 0 and V[1]^2 - V[3]^2 >=0 then printf(`%d, `, n):else fi:od:
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CROSSREFS
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KEYWORD
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nonn,base,fini,full
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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