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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. 1

%I #25 Aug 05 2020 10:13:33

%S 0,1,2,3,4,5,6,7,8,9,10,20,30,40,50,60,70,80,90,101,102,103,104,105,

%T 106,107,108,109,111,121,131,141,151,161,171,181,191,202,203,204,205,

%U 206,207,208,209,212,222,232,242,252,262,272,282,292,303,304,305,306

%N 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.

%C 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.

%H Jinyuan Wang, <a href="/A201982/b201982.txt">Table of n, a(n) for n = 1..145</a>

%H W. Kahan, <a href="https://people.eecs.berkeley.edu/~wkahan/MathH110/Cross.pdf">Computing cross products and rotations in 2 and 3 - dimensional Euclidean spaces</a>, Mathematics and Computer Science Depts. University of California, Berkeley CA 94720 - March 2, 2008.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CrossProduct.html">Cross product</a>

%F 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).

%e 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);

%e 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).

%p 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:

%Y Cf. A002113.

%K nonn,base,fini,full

%O 1,3

%A _Michel Lagneau_, Dec 07 2011

%E Offset changed to 1 by _Jinyuan Wang_, Aug 04 2020

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Last modified March 29 09:32 EDT 2024. Contains 371268 sequences. (Running on oeis4.)