OFFSET
1,1
COMMENTS
-1 < n*r - a(n) < 1 for n >= 1, where r = (5 + sqrt(5))/4.
From Michel Dekking, Dec 28 2017: (Start)
Let (d(n)) be the sequence of first differences: d(n)=a(n+1)-a(n).
CLAIM: d(n) = A108103(n+1) for n=1,2,….
Proof: As a word A287772 = 100110010011001100100110010011... obtained by substituting 0->1, 1->00 in the Fibonacci word F=0100101001001010010100...
This implies that A287772 is a concatenation of 00’s separated by 1’s and 11’s. Moreover, a 0110 occurs iff 1001 occurs in F, and a 010 occurs iff 101 occurs in F. Note also that occurrence of a 00 in A287772 yields a d(n)=1 (and so every other letter in d is a 1), occurrence of a 010 yields a d(n)=2, and occurrence of a 0110 yields a d(n)=3. Since the 1001’s and 101’s occur in 1F according to F itself with 1 prepended (see A001468 and A282162), we must have d(n)=A108103(n+1). (End)
LINKS
Clark Kimberling, Table of n, a(n) for n = 1..10000
MATHEMATICA
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Jun 03 2017
STATUS
approved