A216835 -id:A216835 - cp's OEIS frontend

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).

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

Offset: 1

Yasutoshi Kohmoto, Dec 28 2002

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)

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.

Cf. A000045, A078412, A214094, A214156, A216231, A216275, A216835.

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

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 *)

a(n) = A242603(a(n-1)+a(n-2)). - R. J. Mathar, Mar 13 2024

Corrected by Harvey P. Dale, Jul 23 2012

A224382 Fibonacci-like numbers without positive multiples of 4: a(0) = 0, a(1) = 1, for n>=2, a(n) = a(n-1) + a(n-2) divided by maximal possible power of 4.

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 2, 7, 9, 1, 10, 11, 21, 2, 23, 25, 3, 7, 10, 17, 27, 11, 38, 49, 87, 34, 121, 155, 69, 14, 83, 97, 45, 142, 187, 329, 129, 458, 587, 1045, 102, 1147, 1249, 599, 462, 1061, 1523, 646, 2169, 2815, 1246, 4061, 5307, 2342, 7649, 9991, 4410

Offset: 0

Vladimir Shevelev, Apr 05 2013

nonn

Peter J. C. Moses, Table of n, a(n) for n = 0..9999
B. Avila and T. Khovanova, Free Fibonacci Sequences, arXiv preprint arXiv:1403.4614, 2014 and J. Int. Seq. 17 (2014) # 14.8.5

Cf. A000045, A078414, A214094, A214156, A216231, A216275, A216835.

a[0]:=0; a[1]:=1; a[n_]:=a[n]=#/4^IntegerExponent[#,4]&[(a[n-1]+a[n-2])]; Map[a,Range[0,99]] (* Peter J. C. Moses, Apr 05 2013 *)

A216275 Fibonacci + Goldbach: a(1)=6, a(2)=8 and for n>=3, a(n)=g(a(n-1)) + g(a(n-2)), where for m>=3, g(2m) is the maximal prime p < 2m such that 2*m - p is prime.

Original entry on oeis.org

6, 8, 8, 10, 12, 14, 18, 24, 32, 48, 72, 110, 174, 274, 438, 704, 1134, 1830, 2952, 4762, 7698, 12450, 20128, 32560, 52660, 85168, 137752, 222844, 360564, 583392, 943902, 1527222, 2471074, 3998274, 6469334, 10467566, 16936850, 27404300, 44341050, 71745324

Offset: 1

Vladimir Shevelev, Mar 16 2013

nonn

Conjecture. lim a(n+1)/a(n)=phi as n goes to infinity (phi=golden ratio).

			Let n=6. Since a(4) = 10, a(5) = 12 and g(10) = g(12) = 7, then a(6) = 7 + 7 = 14.

Peter J. C. Moses, Table of n, a(n) for n = 1..1000

Cf. A000045, A002375, A025019, A216835.

a[1] = 6; a[2] = 8; g[n_] := Module[{tmp,k=1}, While[!PrimeQ[n-(tmp=NextPrime[n,-k])], k++]; tmp]; a[n_] := a[n] = g[a[n-1]] + g[a[n-2]]; Table[a[n], {n,1,100}]

For n>=5, a(n) = A216835(n-3) + A216835(n-4).

cp's OEIS Frontend

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).

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A224382 Fibonacci-like numbers without positive multiples of 4: a(0) = 0, a(1) = 1, for n>=2, a(n) = a(n-1) + a(n-2) divided by maximal possible power of 4.

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A216275 Fibonacci + Goldbach: a(1)=6, a(2)=8 and for n>=3, a(n)=g(a(n-1)) + g(a(n-2)), where for m>=3, g(2m) is the maximal prime p < 2m such that 2*m - p is prime.

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cp's OEIS Frontend

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).

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Author

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Programs

Maple

Mathematica

Formula

Extensions

A224382 Fibonacci-like numbers without positive multiples of 4: a(0) = 0, a(1) = 1, for n>=2, a(n) = a(n-1) + a(n-2) divided by maximal possible power of 4.

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A216275 Fibonacci + Goldbach: a(1)=6, a(2)=8 and for n>=3, a(n)=g(a(n-1)) + g(a(n-2)), where for m>=3, g(2*m) is the maximal prime p < 2*m such that 2*m - p is prime.

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A216275 Fibonacci + Goldbach: a(1)=6, a(2)=8 and for n>=3, a(n)=g(a(n-1)) + g(a(n-2)), where for m>=3, g(2m) is the maximal prime p < 2m such that 2*m - p is prime.