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A096016 A modified Fibonacci sequence controlled by a toggle switch. The toggle switch (initial state of 2) flips between 2 and 3 after each reduction. 0

%I

%S 0,1,1,2,3,5,4,3,7,5,4,9,13,11,8,19,27,23,50,73,41,57,98,155,253,136,

%T 389,525,457,982,1439,807,1123,1930,3053,1661,2357,4018,2125,6143,

%U 4134,10277,14411,24688,13033,37721,25377,63098,88475,151573,80016,231589

%N A modified Fibonacci sequence controlled by a toggle switch. The toggle switch (initial state of 2) flips between 2 and 3 after each reduction.

%C Initial state: a(1)=0, a(2)=1, j=2; thereafter a(n) = a(n-1) + a(n-2) and if a(n)>2, j=2 and a(n) == 0 (mod 2) then a(n) = a(n)/j and j=3; if a(n)>3, j=3 and a(n) == 0 (mod 3) then a(n) = a(n)/j and j=2.

%e a(0) = 0 by definition and the initial state of the toggle switch is 2;

%e a(1) = 1 by definition;

%e a(2) = 0+1 = 1;

%e a(3) = 1+1 = 2;

%e a(4) = 1+2 = 3;

%e a(5) = 2+3 = 5;

%e a(6) = 3+5 => 8 => 4 since 8 > 2, therefore 8/2 = 4 and toggle the switch from 2 to 3;

%e a(7) = 5+4 => 9 => 3 since 9 is > 3, therefore 9/3 = 3 and toggle the switch from 3 back to 2;

%e a(8) = 4+3 = 7;

%e a(9) = 3+7 => 10 => 5 since 10 is > 2, therefore 10/2 = 5 and toggle the switch to 3;

%e a(10) = 7+5 => 12 => 4 since 12 is > 3, therefore 12/3 = 4 and toggle the switch to 2;

%e a(11) = 5+4 = 9, cannot reduce by the toggle switch because 9/2 is not an integer;

%e a(12) = 4+9 = 13;

%e a(13) = 9+13 => 22 => 11 and toggle the switch to 3;

%e a(14) = 13+11 => 24 => 8 and set the toggle switch to 2;

%e a(15) = 11+8 = 19, etc.

%t a[0] = 0; a[1] = 1; ts = 0; a[n_] := a[n] = Block[{b = a[n - 1] + a[n - 2]}, If[ IntegerQ[ b / (ts + 2)] && b > 3, b = b/(ts + 2); ts = Mod[ts + 1, 2]]; b]; Table[ a[n], {n, 0, 52}] (* _Robert G. Wilson v_, Jul 21 2004 *)

%K nonn

%O 0,4

%A _Pierre CAMI_, Jul 20 2004

%E Edited by _Robert G. Wilson v_, Jul 21 2004

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Last modified August 5 15:18 EDT 2021. Contains 346472 sequences. (Running on oeis4.)