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.

1, 2, 21, 22, 3, 31, 32, 321, 322, 33, 331, 332, 4, 41, 42, 421, 422, 43, 431, 432, 4321, 4322, 433, 4331, 4332, 44, 441, 442, 4421, 4422, 443, 4431, 4432, 5, 51, 52, 521, 522, 53, 531, 532, 5321, 5322, 533, 5331, 5332, 54, 541, 542, 5421, 5422, 543, 5431, 5432

Offset: 1

Author

Jonas Hoeglund (firefly(AT)firefly.nu), Dec 11 2009

Keywords

• nonn
• base

Comments

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.

See the Zeckendorf expansion of n (A035517) for a similar expansion. - N. J. A. Sloane, Dec 12 2009

Programs

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; }

Maple

Contribution from R. J. Mathar, Oct 23 2010: (Start)
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:
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:
seq(A171549(n),n=1..80) ; (End)

Extensions

Extended by R. J. Mathar, Oct 23 2010