The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A078414 a(n) = (a(n-1)+a(n-2))/7^k, where 7^k is the highest power of 7 dividing a(n-1)+a(n-2). 8
 1, 1, 2, 3, 5, 8, 13, 3, 16, 19, 5, 24, 29, 53, 82, 135, 31, 166, 197, 363, 80, 443, 523, 138, 661, 799, 1460, 2259, 3719, 122, 3841, 3963, 7804, 1681, 1355, 3036, 4391, 1061, 5452, 6513, 11965, 18478, 4349, 3261, 7610, 1553, 187, 1740, 1927, 3667, 5594, 27 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS From Vladimir Shevelev, Apr 01 2013; edited by Danny Rorabaugh, Feb 19 2016: (Start) If we consider Fibonacci-like numbers {F_p(n)} without positive multiples of p, where p is a fixed prime, then {F_2(n)} has period of length 1, {F_3(n)} has period of length 3, {F_5(n)} has period of length 6. This sequence is the first which does not have a trivial period and, probably, even is non-periodic. An open question: Is this sequence bounded? Consider Fibonacci-like sequences without multiples of several primes, defined analogously: e.g., for {F_(p,q)(n)}, a(0)=0, a(1)=1, for n>=2, a(n)=a(n-1)+a(n-2) divided by the maximal possible powers of p and q. Problem: For what sets of primes is the corresponding Fibonacci-like sequence without multiples of these primes periodic? Examples: sequence {F_(7,11,13)(n)} has period of length 12: 0, 1, 1, 2, 3, 5, 8, 1, 9, 10, 19, 29, 48, 1, 1, 2, 3, 5,...; {F_(11,13,19)(n)} has period of length 9; {F_(13,19,23)(n)} has period of length 12; {F_(17,19,23,29)(n)} has period of length 15; {F_(19,23,31,53,59,89)(n)} has period of length 24; {F_(23,29,73,233)(n)} has period of length 18. Don Reble noted that lengths of all such periods could only be multiples of 3 because every Fibonacci-like sequence considered here modulo 2 has the form 0,1,1,0,1,1,... . (End) LINKS Alois P. Heinz, Table of n, a(n) for n = 1..4000 B. Avila, T. Khovanova, Free Fibonacci Sequences, arXiv preprint arXiv:1403.4614 [math.NT], 2014 and J. Int. Seq. 17 (2014) # 14.8.5. FORMULA a(n) = A242603(a(n-1)+a(n-2)). - R. J. Mathar, Mar 13 2024 MAPLE a:= proc(n) option remember; local t, j; if n<3 then 1 else t:= a(n-1)+a(n-2); while irem(t, 7, 'j')=0 do t:=j od; t fi end: seq(a(n), n=1..100); # Alois P. Heinz, Jul 25 2012 MATHEMATICA nxt[{a_, b_}]:=Module[{n=IntegerExponent[a+b, 7]}, {b, (a+b)/7^n}]; Transpose[ NestList[nxt, {1, 1}, 60]][[1]] (* Harvey P. Dale, Jul 23 2012 *) CROSSREFS Cf. A000045, A078412, A214094, A214156, A216231, A216275, A216835. Sequence in context: A072123 A135102 A214156 * A254056 A238948 A336716 Adjacent sequences: A078411 A078412 A078413 * A078415 A078416 A078417 KEYWORD nonn,easy AUTHOR Yasutoshi Kohmoto, Dec 28 2002 EXTENSIONS Corrected by Harvey P. Dale, Jul 23 2012 STATUS approved

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

Last modified June 17 15:57 EDT 2024. Contains 373463 sequences. (Running on oeis4.)