The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation. Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 56th year, we are closing in on 350,000 sequences, and we’ve crossed 9,700 citations (which often say “discovered thanks to the OEIS”). Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A105774 A "fractal" transform of the Fibonacci numbers: a(1)=1; then if F(n) < k <= F(n+1), a(k) = F(n+1) - a(k - F(n)) where F(n) = A000045(n). 6
 1, 1, 2, 4, 4, 7, 7, 6, 12, 12, 11, 9, 9, 20, 20, 19, 17, 17, 14, 14, 15, 33, 33, 32, 30, 30, 27, 27, 28, 22, 22, 23, 25, 25, 54, 54, 53, 51, 51, 48, 48, 49, 43, 43, 44, 46, 46, 35, 35, 36, 38, 38, 41, 41, 40, 88, 88, 87, 85, 85, 82, 82, 83, 77, 77, 78, 80, 80, 69, 69, 70, 72, 72 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS Let tau = (1+sqrt(5))/2; then the missing numbers 3,5,8,10,13,16,18,21,... are given by round(tau^2*k) for k > 0 (A004937). Indices n such that a(n) = a(n+1) are given by floor(tau^2*k) - 1 for k > 0 (A003622). Numbers n such that a(n) differs from a(n+1) are given by floor(tau*k+1/tau) for k > 0 (A022342). Indices n giving isolated terms (a(n) differs from a(n-1) and a(n+1)) are given by floor(tau*floor(tau^2*k)) for k > 0 (A003623). Remove 0's from the first differences of sorted values; then you get a version of the infinite Fibonacci word (A001468). I.e., sorted values are 1,1,2,4,4,6,7,7,9,9,11,12,12,..., first differences are 0,1,2,0,2,1,0,2,0,2,1,0,2,0,1,...; removing 0's gives 1,2,2,1,2,2,1,2,1,2,2,1,2,1,2,2,1,2,... #{ k : a(k)=k}=infty. LINKS Antti Karttunen, Table of n, a(n) for n = 1..10946 FORMULA a(A000045(n)) = A006498(n-1) for n >= 1. - Typo corrected by Antti Karttunen, Mar 17 2017 limsup a(n)/n = tau and liminf a(n)/n = 1/tau where tau = (1+sqrt(5))/2. a(n) mod 2 = A085002(n) - Benoit Cloitre, May 10 2005 a(1) = 1; for n > 1, a(n) = A000045(2+A072649(n-1)) - a(n-A000045(1 + A072649(n-1))). - Antti Karttunen, Mar 17 2017 EXAMPLE For 1 = F(2) < k <= F(3) = 2 the rule gives a(2) = 2 - a(1) = 1 ... if 5 = F(5) < k <= F(6) = 8 the rule forces a(6) = 8 - a(6-5) = 8 - a(1) = 7; a(7) = 8 - a(2) = 7; a(8) = 8 - a(3) = 6. PROG (Scheme, with memoization-macro definec) (definec (A105774 n) (if (= 1 n) n (- (A000045 (+ 2 (A072649 (- n 1)))) (A105774 (- n (A000045 (+ 1 (A072649 (- n 1))))))))) ;; Antti Karttunen, Mar 17 2017 CROSSREFS Cf. A000045, A001468, A003622, A003623, A004937, A006498, A022342, A072649, A085002, A105669, A105670, A105672, A093347, A093348, A283766. Sequence in context: A274593 A103622 A328853 * A130805 A023831 A233272 Adjacent sequences:  A105771 A105772 A105773 * A105775 A105776 A105777 KEYWORD nonn AUTHOR Benoit Cloitre, May 04 2005 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 9 01:52 EST 2021. Contains 349625 sequences. (Running on oeis4.)