A284282 a(n) = the number k such that A030067(2k-1) = n, or 0 if n does not occur in the semi-Fibonacci sequence A030067.

0, 1, 2, 3, 0, 4, 5, 0, 0, 6, 0, 7, 0, 0, 0, 0, 8, 9, 0, 0, 0, 0, 0, 10, 0, 0, 11, 0, 0, 0, 0, 0, 0, 0, 0, 12, 0, 13, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 14, 0, 0, 0, 0, 15, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 16, 17, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 18

Offset: 0

Author

M. F. Hasler, Mar 24 2017

Keywords

• nonn

Comments

Otherwise said, a(n) = round(m/2) = (m+1)/2, where m is the smallest index such that A030067(m) = n.

Any integer n which occurs in A030067 first occurs as an odd-indexed term A030067(2k-1) = A030068(k-1), and thereafter at indices (2k-1)*2^j, j=1,2,3,... (Both of these statements follow immediately from the definition of even-indexed terms of A030067.)

It is easy to see that no n can occur a second time as an odd-indexed term in A030067. This follows from the definition of these terms A030067(2k+1) = A030067(2k-1) + A030067(k), which shows that the subsequence of odd-indexed terms (A030068) is strictly increasing, and therefore equal to the range (or: set) of all semi-Fibonacci numbers.

Setting all nonzero terms to 1, this sequence is the characteristic function of A030068 (up to the offset).

Crossrefs

Cf. A030067 (semi-Fibonacci sequence), A030068 (bisection of odd-indexed terms, also equal to the range = set of all possible values or semi-Fibonacci numbers).

Programs

Mathematica

a[n_] := a[n] = Which[n == 1, 1, EvenQ@ n, a[n/2], True, a[n - 1] + a[n - 2]]; With[{nn = 87}, Function[s, Function[t, {0}~Join~ReplacePart[t, Map[# -> First@ Lookup[s, #] &, TakeWhile[Keys@ s, # <= nn &]]]]@ ConstantArray[0, nn]]@ PositionIndex@ Array[a[2 # - 1] &, 10^3]] (* Michael De Vlieger, Mar 25 2017, Version 10, after Jean-François Alcover at A030067 *)

PARI

A284282(n)=setsearch(A030068_vec,n) \\ Use, e.g., A030068(100) to compute the global variable A030068_vec far enough for n <= 22880. - M. F. Hasler, Mar 25 2017