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%I #3 Mar 30 2012 17:30:30
%S 0,1,1,2,3,5,4,3,7,2,3,5,2,1,1,1,1,2,3,1,4,5,3,4,7,11,6,17,23,10,3,13,
%T 4,1,1,2,1,3,1,2,3,5,4,3,1,1,2,1,1,2,3,1,2,3,5,4,3,7,1,4,1,5,1,2,1,3,
%U 1,1,2,3,1,2,1,3,4,1,5,3,4,7,11,6,17,1,6,1,7,2,3,1,1,2,1,3,2,5,1,1,1,2,1
%N Improperly Reduced Fibonacci Sequence: begin with a(0) = 0, a(1) = 1 and a(n) = [ a(n-1) + a(n-2) ] / a(k). a(k) is the first (not necessarily the greatest) term including 1 which divides a(n-1) + a(n-2) not previously used.
%e a(6) = 4 since a(4) + a(5) = 3 + 5 which equals 8 but is divisible by a(3) which equals 2. a(3) is no longer available for future consideration as a divisor.
%t y = 0; c = l = i = z = 1; d = {1}; Print[ 0 ]; Print[ 1 ]; Do[ x = y + z; c++; j = 1; While[ ! IntegerQ[ x/d[ [ j ] ] ] && j <= i, j++ ]; If[ j > i, d = Append[ d, x ]; i++, x = x / d[ [ j ] ]; d = Delete[ d, j ]; d = Append[ d, x ] ]; Print[ x ]; y = z; z = x, {n, 1, 100} ]
%o (BASIC?): 10 ! Improperly Reduced Fibonacci Sequence. 20 Option Base 1 @ Dim F(1),S$[65530] 30 Z=1 @ Y=0 @ F(1)=1 40 S$='0,1,' @ C=1 @ D=1 100 C=C+1 110 N=Y+Z 120 For I=1 To D 140 If Not Rmd(N,F(I)) Then 160 150 Next I @ D=D+1 @ Dim F(D) @ Goto 180 160 N=N/F(I) 170 For I=I To D-1 @ F(I)=F(I+1) @ Next I 180 F(D)=N @ S$=S$&Str$(N)&',' 190 Y=Z @ Z=N 200 Goto 100
%Y Cf. A000045.
%K nonn
%O 0,4
%A _Robert G. Wilson v_, Jan 15 2001
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