A116470 -id:A116470 - cp's OEIS frontend

A272912 Difference sequence of the sequence A116470 of all distinct Fibonacci numbers and Lucas numbers (A000032).

1, 1, 1, 1, 2, 1, 3, 2, 5, 3, 8, 5, 13, 8, 21, 13, 34, 21, 55, 34, 89, 55, 144, 89, 233, 144, 377, 233, 610, 377, 987, 610, 1597, 987, 2584, 1597, 4181, 2584, 6765, 4181, 10946, 6765, 17711, 10946, 28657, 17711, 46368, 28657, 75025, 46368, 121393, 75025

Offset: 1

Clark Kimberling, May 10 2016

Every term is a Fibonacci number (A000045).

			A116470 = (1, 2, 3, 4, 5, 7, 8, 11, 13, 18, 21, 29, 34, 47, 55, 76,...), so that (a(n)) = (1,1,1,1,2,1,3,2,5,3,8,5,13,8,12,...).

Clark Kimberling, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,1,0,1).

Cf. A000045, A000032, A116470, A053602, A123231.

u = Table[Fibonacci[n], {n, 1, 200}]; v = Table[LucasL[n], {n, 1, 200}];
Take[Differences[Union[u, v]], 100]

Vec(x*(1+x-x^5)/(1-x^2-x^4) + O(x^50)) \\ Colin Barker, May 10 2016

From Colin Barker, May 10 2016: (Start)
a(n) = a(n-2)+a(n-4) for n>4.
G.f.: x*(1+x-x^5) / (1-x^2-x^4).
(End)
a(n) = A053602(n-2), n>2. - R. J. Mathar, May 20 2016
a(n) = A123231(n-3), n>3. - Georg Fischer, Oct 23 2018

A214973 Number of terms in greedy representation of n using Fibonacci and Lucas numbers.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 2, 2, 3, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 3, 2, 2, 3, 3, 2, 3, 1, 2, 2, 2, 2, 2, 3, 2, 1, 2, 2, 2, 2, 2, 3, 2, 2, 3, 3, 2, 3, 2, 3, 3, 3, 3, 2, 3, 3, 1, 2, 2, 2, 2, 2, 3, 2, 2, 3, 3, 2, 3, 1, 2, 2, 2, 2, 2, 3, 2, 2, 3, 3

Offset: 1

Clark Kimberling, Oct 20 2012

nonn

Consider the sequence b = A116470 consisting of all the Fibonacci numbers and Lucas numbers.  For n>=0, let k(1) be the greatest k in the basis b = {b(k)} such that b(k) <= n, let k(2) be the greatest k in b such that k <= n-b(k(1)), and so on, resulting in the greedy b-representation (to be abbreviated as "rep") of n.  For comparison with the rep using A000045 as basis (called Zeckendorf, or Fibonacci, rep) and also with the rep using A000032 as basis (called the Lucas rep), it is natural to ask this:  what terms in b can possibly follow a given term b(k)?  The answer follows.
If b(k) < 21 = b(11), the terms that can follow b(k) are easily found and not recorded here.  Otherwise, if k is odd, then b(k) can be followed by b(k-i) for some i>=5; if k is even, then b(k) can be followed by b(k-i) for some i>=8.  In Zeckendorf and Lucas reps, the "lag" is k-i for i>=2.
Conjecture: a(A049651(n)) = n and this is the first instance of n in the sequence for all n > 0. In other words, apart from its initial term, A049651 is the RECORDS transform of this sequence. - Charles R Greathouse IV, Oct 14 2021

			Let F, L, U denote the Fibonacci (aka Zeckendorf), Lucas, and greedy F-union-L representations.  Then 45 = 34+8+3 (F) = 29+11+4+1 (L) = 34+11 (U), which shows that a(45) = 2 and that the U representation of 45 requires fewer terms than the others; 45 is the least number having this property.

Clark Kimberling, Table of n, a(n) for n = 1..10000
Jon Maiga, Computer-generated formulas for A214973, Sequence Machine.

Cf. A000032, A000045, A007895, A116470.

s = Reverse[Union[Flatten[Table[{Fibonacci[n + 1], LucasL[n - 1]}, {n, 1, 22}]]]]; Map[Length[Select[Reap[FoldList[(Sow[Quotient[#1, #2]]; Mod[#1, #2]) &, #, s]]
[[2, 1]], # > 0 &]] &, Range[120]]
(* Peter J. C. Moses, Oct 18 2012 *)

w(n)=if(n%2, fibonacci(n\2+3), fibonacci(n\2) + fibonacci(n\2+2));
k(n)=if(n<9, return(if(n==6,5,n))); for(i=8,n, if(w(i)>n, return(w(i-1))));
a(n)=my(s); while(n, n-=k(n); s++); s; \\ Charles R Greathouse IV, Oct 14 2021

Conjecture: a(n) = A329320(A048679(n)) for n > 0 (noticed by Sequence Machine). - Mikhail Kurkov, Oct 13 2021

A115339 a(2n-1)=F(n+1), a(2n)=L(n), where F(n) and L(n) are the Fibonacci and the Lucas sequences.

Original entry on oeis.org

1, 1, 2, 3, 3, 4, 5, 7, 8, 11, 13, 18, 21, 29, 34, 47, 55, 76, 89, 123, 144, 199, 233, 322, 377, 521, 610, 843, 987, 1364, 1597, 2207, 2584, 3571, 4181, 5778, 6765, 9349, 10946, 15127, 17711, 24476, 28657, 39603, 46368, 64079, 75025, 103682, 121393, 167761

Offset: 1

Giuseppe Coppoletta, Mar 06 2006

Alternate Fibonacci and Lucas sequence respecting their natural order.
See A116470 for an essentially identical sequence.
The ratio a(n+1)/a(n) increasingly approximates two constants: (5-sqrt(5))/2 (A094874) and (5+3*sqrt(5))/10 (A176015) according to whether n is odd or even. - Davide Rotondo, Oct 27 2024

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Fibonacci Number
Eric Weisstein's World of Mathematics, Lucas Number.
Index entries for linear recurrences with constant coefficients, signature (0,1,0,1).

Cf. A000045, A000032.
Cf. A000930.
Cf. A094874, A176015.

a115339 n = a115339_list !! (n-1)
a115339_list = [1, 1, 2, 3] ++
               zipWith (+) a115339_list (drop 2 a115339_list)
-- Reinhard Zumkeller, Aug 03 2013

f[n_] := If[OddQ@n, Fibonacci[(n + 3)/2], Fibonacci[n/2 - 1] + Fibonacci[n/2 + 1]]; Array[f, 50] (* Robert G. Wilson v, Apr 29 2006 *)

x='x+O('x^50); Vec(x*(-1-x-x^2-2*x^3)/(-1+x^2+x^4)) \\ G. C. Greubel, Apr 27 2017

a(n+2) = a(n) + a(n-2).

G.f.: x*( -1-x-x^2-2*x^3 ) / ( -1+x^2+x^4 ). - R. J. Mathar, Mar 08 2011

More terms from Robert G. Wilson v, Apr 29 2006

A355324 Lower midsequence of the Fibonacci numbers (1,2,3,5,8,...) and Lucas numbers (1,3,4,7,11,...); see Comments.

Original entry on oeis.org

1, 2, 3, 6, 9, 15, 25, 40, 65, 106, 171, 277, 449, 726, 1175, 1902, 3077, 4979, 8057, 13036, 21093, 34130, 55223, 89353, 144577, 233930, 378507, 612438, 990945, 1603383, 2594329, 4197712, 6792041, 10989754, 17781795, 28771549, 46553345, 75324894, 121878239

Offset: 0

Clark Kimberling, Jul 16 2022

Suppose that s = (s(n)) and t = (t(n)) are integer sequences. The lower midsequence, m = m(s,t), of s and t is defined by m(n) = floor((s(n) + t(n))/2). The upper midsequence, M = M(s,t), is defined by M(n) = ceiling((s(n) + t(n))/2).

Here, s(n) = F(n+2) and t(n) = L(n+1), for n >= 0, where F = A000045 (Fibonacci numbers) and L = A000032 (Lucas numbers).

			a(0) = 1 = floor((1+1)/2);
a(1) = 2 = floor((2+3)/2);
a(2) = 3 = floor((3+4)/2).
The Fibonacci and Lucas numbers are interspersed:
1 < 2 < 3 < 4 < 5 < 7 < 8 < 11 < 13 < 18 < 21 < 29 < ...
The midsequences m and M intersperse the ordered union of the Fibonacci and Lucas sequences, A116470, as indicated by the following table:
   F    m    M    L
   1    1    1    1
   2    2    3    3
   3    3    4    4
   5    6    6    7
   8    9   10   11
  13   15   16   18
  21   25   25   29

Index entries for linear recurrences with constant coefficients, signature (1,1,1,-1,-1).

Cf. A000032, A000045, A116470, A355325.

Table[Floor[(LucasL[n + 1] + Fibonacci[n + 2])/2], {n, 0, 50}]    (* A355324 *)
Table[Ceiling[(LucasL[n + 1] + Fibonacci[n + 2])/2], {n, 0, 50}]  (* A355325 *)

from sympy import fibonacci, lucas
def A355324(n): return fibonacci(n+2)+lucas(n+1)>>1 # Chai Wah Wu, Aug 08 2022

a(n) = floor((A000045(n+2) + A000032(n+1))/2).
a(n) = a(n-1) + a(n-2) + a(n-3) - a(n-4) - a(n-5) for n >= 5.
G.f.: (1 + x - x^4)/(1 - x - x^2 - x^3 + x^4 + x^5).
G.f.: ((1 + x - x^4)/((-1 + x) (-1 + x + x^2) (1 + x + x^2))).
a(n) = (3*((5 - 4*sqrt(5))*(1 - sqrt(5))^n + (1 + sqrt(5))^n*(5 + 4*sqrt(5)))/2^n + 10*(cos(2*n*Pi/3) - 1))/30. - Stefano Spezia, Jul 17 2022

A355325 Upper midsequence of the Fibonacci numbers (1,2,3,5,8,...) and Lucas numbers (1,3,4,7,11,...); see Comments.

Original entry on oeis.org

1, 3, 4, 6, 10, 16, 25, 41, 66, 106, 172, 278, 449, 727, 1176, 1902, 3078, 4980, 8057, 13037, 21094, 34130, 55224, 89354, 144577, 233931, 378508, 612438, 990946, 1603384, 2594329, 4197713, 6792042, 10989754, 17781796, 28771550, 46553345, 75324895, 121878240

Offset: 0

Clark Kimberling, Jul 16 2022

Suppose that s = (s(n)) and t = (t(n)) are integer sequences. The lower midsequence, m = m(s,t), of s and t is defined by m(n) = floor((s(n) + t(n))/2). The upper midsequence, M = M(s,t), is defined by M(n) = ceiling((s(n) + t(n))/2).

Here, s(n) = F(n+2) and t(n) = L(n+1), for n >= 0, where F = A000045 (Fibonacci numbers) and L = A000032 (Lucas numbers).

			a(0) = 1 = ceiling((1+1)/2);
a(1) = 3 = ceiling((2+3)/2);
a(2) = 4 = ceiling((3+4)/2).
The Fibonacci and Lucas numbers are interspersed:
1 < 2 < 3 < 4 < 5 < 7 < 8 < 11 < 13 < 18 < 21 < 29 < ...
The midsequences m and M intersperse the ordered union of the Fibonacci and Lucas sequences, A116470, as indicated by the following table:
   F    m    M    L
   1    1    1    1
   2    2    3    3
   3    3    4    4
   5    6    6    7
   8    9   10   11
  13   15   16   18
  21   25   25   29

Index entries for linear recurrences with constant coefficients, signature (1,1,1,-1,-1).

Cf. A000032, A000045, A116470, A355324.

Table[Floor[(LucasL[n + 1] + Fibonacci[n + 2])/2], {n, 0, 50}]    (* A355324 *)
Table[Ceiling[(LucasL[n + 1] + Fibonacci[n + 2])/2], {n, 0, 50}]  (* A355325 *)

a(n) = ceiling((A000045(n+2) + A000032(n+1))/2).
a(n) = a(n-1) + a(n-2) + a(n-3) - a(n-4) - a(n-5) for n >= 5.
G.f.: (1 + 2 x - 2 x^3 - 2 x^4)/(1 - x - x^2 - x^3 + x^4 + x^5).
G.f.: ((1 + 2 x - 2 x^3 - 2 x^4)/((-1 + x) (-1 + x + x^2) (1 + x + x^2))).
a(n) = (10 + 3*((5 - 4*sqrt(5))*(1 - sqrt(5))^n + (1 + sqrt(5))^n*(5 + 4*sqrt(5)))/2^n - 10*cos(2*n*Pi/3))/30. - Stefano Spezia, Jul 17 2022

A378698 Number of compositions of n into parts whose sizes are Fibonacci or Lucas numbers.

Original entry on oeis.org

1, 1, 2, 4, 8, 16, 31, 62, 123, 243, 481, 953, 1887, 3737, 7399, 14652, 29014, 57452, 113767, 225279, 446095, 883352, 1749201, 3463746, 6858864, 13581833, 26894570, 53256275, 105457382, 208825335, 413513204, 818833458, 1621443338, 3210760963, 6357907009

Offset: 0

Davide Rotondo, Dec 04 2024

nonn

a(n+1)/a(n) approximate the constant r = 1.9801869...

Cf. A000032, A000045, A116470.

Cf. A076739, A288039.

A116470(n) = if(n<6, n, if(n%2, fibonacci(n\2+3), fibonacci(n\2)+fibonacci(n\2+2)))
a(max_n) = {Vec(1/(1+sum(k=1,max_n-1, -1*x^A116470(k)))+O(x^max_n)); } \\ Thomas Scheuerle, Dec 04 2024

G.f.: 1/(1 - Sum_{k>=1} x^A116470(k)). - Thomas Scheuerle, Dec 04 2024

cp's OEIS Frontend

A272912 Difference sequence of the sequence A116470 of all distinct Fibonacci numbers and Lucas numbers (A000032).

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A214973 Number of terms in greedy representation of n using Fibonacci and Lucas numbers.

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A115339 a(2n-1)=F(n+1), a(2n)=L(n), where F(n) and L(n) are the Fibonacci and the Lucas sequences.

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A355324 Lower midsequence of the Fibonacci numbers (1,2,3,5,8,...) and Lucas numbers (1,3,4,7,11,...); see Comments.

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A355325 Upper midsequence of the Fibonacci numbers (1,2,3,5,8,...) and Lucas numbers (1,3,4,7,11,...); see Comments.

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A378698 Number of compositions of n into parts whose sizes are Fibonacci or Lucas numbers.

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