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A171549 a(n) is the n-th "strange number", where "strange numbers" are defined as follows: using every other Fibonacci number (starting with offset 1), the shortest way to add up these Fibonacci numbers so that the sum equals n, where the digits are indices in the "every other Fibonacci number" sequence.

%I A171549 #10 May 30 2022 16:32:49
%S A171549 1,2,21,22,3,31,32,321,322,33,331,332,4,41,42,421,422,43,431,432,4321,
%T A171549 4322,433,4331,4332,44,441,442,4421,4422,443,4431,4432,5,51,52,521,
%U A171549 522,53,531,532,5321,5322,533,5331,5332,54,541,542,5421,5422,543,5431,5432
%N A171549 a(n) is the n-th "strange number", where "strange numbers" are defined as follows: using every other Fibonacci number (starting with offset 1), the shortest way to add up these Fibonacci numbers so that the sum equals n, where the digits are indices in the "every other Fibonacci number" sequence.
%C A171549 We can also find these numbers as follows: We have an alphabet {1, 2, 3, 4, 5, ...}, and a number known as the "limit", initially set to 1. to describe the first number, we use the first character of the alphabet: 1 to describe 2, we have to increase the limit and define a new character: 2 to describe 3, the limit is now 2, so we may use two characters. 2+1 = 3, therefore: 21 to describe 4, we use 22 to describe 5, we need a new character: 3. the limit is now increased to 3. etc.
%C A171549 See the Zeckendorf expansion of n (A035517) for a similar expansion. - _N. J. A. Sloane_, Dec 12 2009
%p A171549 Contribution from _R. J. Mathar_, Oct 23 2010: (Start)
%p A171549 read("transforms") ; A130234 := proc(n) local m,N,a,i ; for m from 0 do if combinat[fibonacci](m) >= n then break ; end if; end do; m ; end proc:
%p A171549 A171549 := proc(n) local m,N,a,i ; m := A130234(n) ; if type(m,'odd') then m := m+1 ; end if; N := n ; a := 0 ; while N >0 do for i from m to 1 by -1 do if N >= combinat[fibonacci](2*i-1) then N := N- combinat[fibonacci](2*i-1) ; a := digcat2(a,i) ; break ; end if; end do: end do: return a; end proc:
%p A171549 seq(A171549(n),n=1..80) ; (End)
%o A171549 (Java) // fib(n) gives fibonacci number n. public static String S(int n) { int max; for (max = 0; fib(max) < n; max += 2); int num = n; String out = ""; while (num > 0) { for (int i = max; i>0; i--) if (num >= fib(2*i-1)) { num -= fib(2*i-1); out += (char)('0' + i); break; } } return out; }
%K A171549 nonn,base
%O A171549 1,2
%A A171549 Jonas Hoeglund (firefly(AT)firefly.nu), Dec 11 2009
%E A171549 Extended by _R. J. Mathar_, Oct 23 2010