Search: id:A118229
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%I A118229
%S A118229 1,1,1,1,0,1,1,1,1,1,1,0,0,0,1,1,0,0,1,1,1,1,0,1,0,1,0,1,1,0,2,1,0,0,1,
%T A118229 1,1,0,0,0,1,0,1,0,1,1,0,1,1,0,1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,1,0,1,0,
%U A118229 0,0,1,0,0,1,1,1,3,0,2,0,2,0,2,0,1,0,1,0,1,3,0,1,0,3,0,1,1,1,0,0,0,1,1
%V A118229 1,-1,1,-1,0,1,1,-1,-1,1,-1,0,0,0,1,1,0,0,-1,-1,1,1,0,-1,0,-1,0,1,-1,0,
               2,-1,0,0,-1,1,
%W A118229 -1,0,0,0,1,0,-1,0,1,1,0,-1,1,0,-1,1,-1,-1,1,-1,0,1,0,0,0,-1,0,0,0,1,1,
               0,-1,0,0,0,1,0,
%X A118229 0,-1,-1,1,3,0,-2,0,-2,0,2,0,-1,0,-1,0,1,-3,0,1,0,3,0,-1,-1,1,0,0,0,-1,
               1
%N A118229 Triangle, read by rows, equal to the matrix inverse of triangle A054431; 
               the inverse transformation that obtains {a(n)} from b(n) = sum{1<=k<=n, 
               GCD(k,n)=1} a(k).
%C A118229 Column 1 is A096433. Column 2 = [0,1,0,-1,0,0,0,...(zero for n>4)]. Column 
               3 is A118230.
%H A118229 Leroy Quet, <a href="http://www.prism-of-spirals.net/">Home Page</a> 
               (listed in lieu of email address)
%F A118229 For column k>1: Sum_{i=2..n, gcd(n,i)=1} T(i,k) = 1 when n=k+1, 0 elsewhere; 
               for column k=1: Sum_{i=2..n, gcd(n,i)=1} T(i,1) = 1 when n=1 or 2, 
               0 elsewhere.
%e A118229 Describes a sequence transformation as follows.
%e A118229 Say we have the arbitrary sequence {a(k)}.
%e A118229 We define {b(k)}, based on {a(k)}, by:
%e A118229 b(n) = sum{1<=k<=n, GCD(k,n)=1} a(k).
%e A118229 So given {b(k)} (which must have b(1) = b(2)), how do we get the sequence 
               {a(k)}?
%e A118229 If a(n) = sum{k>=2} b(k) * T(n,k), then there is a triangular array {T(n,
               k)} which begins:
%e A118229 1;
%e A118229 -1, 1;
%e A118229 -1, 0, 1;
%e A118229 1,-1,-1, 1;
%e A118229 -1, 0, 0, 0, 1;
%e A118229 1, 0, 0,-1,-1, 1;
%e A118229 1, 0,-1, 0,-1, 0, 1;
%e A118229 -1, 0, 2,-1, 0, 0,-1, 1;
%e A118229 -1, 0, 0, 0, 1, 0,-1, 0, 1;
%e A118229 1, 0,-1, 1, 0,-1, 1,-1,-1, 1;
%e A118229 -1, 0, 1, 0, 0, 0,-1, 0, 0, 0, 1;
%e A118229 1, 0,-1, 0, 0, 0, 1, 0, 0,-1,-1, 1;
%e A118229 3, 0,-2, 0,-2, 0, 2, 0,-1, 0,-1, 0, 1;
%e A118229 -3, 0, 1, 0, 3, 0,-1,-1, 1, 0, 0, 0,-1, 1; ...
%o A118229 (PARI) {T(n,k)=if(n<k|k<0,0,(matrix(n,n,r,c,if(r>=c,if(gcd(r-c+1,c)==1,
               1,0)))^-1)[n,k])}
%Y A118229 Cf. A054431 (matrix inverse), A096433 (column 1), A118230 (column 3).
%Y A118229 Sequence in context: A056929 A151692 A115201 this_sequence A117201 A060953 
               A082858
%Y A118229 Adjacent sequences: A118226 A118227 A118228 this_sequence A118230 A118231 
               A118232
%K A118229 sign,tabl
%O A118229 1,31
%A A118229 Leroy Quet, Paul D. Hanna (pauldhanna(AT)juno.com), Apr 16 2006

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