

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 3dimensional Euclidean space. Sequence gives numbers n such that the components u, v, w are > = 0.


1



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

1,3


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.


LINKS

Jinyuan Wang, Table of n, a(n) for n = 1..145
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.
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*bb*c, c^2a^2, a*bb*c).


EXAMPLE

2 is in the sequence because (0,0,2)^(2,0,0) = (0*02*0, 2*20*0, 0*00*2) = (0,4,0);
509 is in the sequence because (5,0,9)^(9,0,5) = (0*50*9, 9*95*5, 5*00*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,base,fini,full


AUTHOR

Michel Lagneau, Dec 07 2011


EXTENSIONS

Offset changed to 1 by Jinyuan Wang, Aug 04 2020


STATUS

approved



