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A201982 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. 0
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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.

REFERENCES

W. Kahan, Computing cross products and rotations in 2 and 3 - dimensional Euclidean spaces, Mathematics and Computer Science Depts. University of California, Berkeley CA 94720 - March 2, 2008.

LINKS

Table of n, a(n) for n=0..57.

Eric Weisstein's World of Mathematics, Cross product

FORMULA

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).

EXAMPLE

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).

MAPLE

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:

CROSSREFS

Cf. A002113.

Sequence in context: A071061 A037124 A273737 * A166508 A223080 A248651

Adjacent sequences:  A201979 A201980 A201981 * A201983 A201984 A201985

KEYWORD

nonn,fini,base

AUTHOR

Michel Lagneau, Dec 07 2011

STATUS

approved

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Last modified November 21 04:41 EST 2019. Contains 329350 sequences. (Running on oeis4.)