A115339 -id:A115339 - cp's OEIS frontend

A116470 All distinct Fibonacci and Lucas numbers.

0, 1, 2, 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: 0

Alexander Adamchuk, Aug 13 2006

See A115339 for an essentially identical sequence.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0,1,0,1).

Union of A000045 and A000032.

Cf. A288219 (even bisection).

import Data.List (transpose)
a116470 n = a116470_list !! n
a116470_list = 0 : 1 : 2 : concat
               (transpose [drop 4 a000045_list, drop 3 a000032_list])
-- Reinhard Zumkeller, Aug 03 2013

FL := (n, a, b) -> hypergeom([a - n/2, b - n/2], [1 - n], -4):
a := n -> `if`(n < 3, n, FL((n + 2 + 3*irem(n, 2))/2, irem(n, 2), 1/2)):
seq(simplify(a(n)), n=0..52); # Peter Luschny, Sep 03 2019

CoefficientList[Series[-x*(x^2 + x + 1)*(x^3 + x + 1)/(-1 + x^4 + x^2), {x, 0, 50}], x] (* G. C. Greubel, Dec 21 2017 *)
With[{nn=50},Select[Union[Join[LucasL[Range[0,nn]],Fibonacci[Range[0,nn]]]],#<=200000&]] (* Harvey P. Dale, Jul 05 2019 *)

my(x='x+O('x^50)); concat([0], Vec(-x*(x^2+x+1)*(x^3+x+1)/( -1+x^4 +x^2))) \\ G. C. Greubel, Dec 21 2017

a(n)=if(n<6, n, if(n%2, fibonacci(n\2+3), fibonacci(n\2)+fibonacci(n\2+2))) \\ Charles R Greathouse IV, Oct 14 2021

a(0) = 0, a(1) = 1, a(2) = 2, a(3) = 3, a(4) = 4, a(5) = 5, a(6) = 7, a(n) = a(n-2) + a(n-4) for n>6.
a(2*n) = Lucas(n+1) = Fibonacci(n) + Fibonacci(n+2) for n>1.
a(2*n+1) = Fibonacci(n+3) for n>2.
G.f.: -x*(x^2+x+1)*(x^3+x+1)/(-1+x^4+x^2). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 12 2009
a(n) = FL((n + 2 + 3*(n mod 2))/2, n mod 2, 1/2) for n >= 3. Here FL(n, a, b) = hypergeom([a - n/2, b - n/2], [1 - n], -4). - Peter Luschny, Sep 03 2019

A180236 a(n) = a(n-2)+a(n-4); a(1)=a(4)=101, a(2)=a(3)=10.

Original entry on oeis.org

101, 10, 10, 101, 111, 111, 121, 212, 232, 323, 353, 535, 585, 858, 938, 1393, 1523, 2251, 2461, 3644, 3984, 5895, 6445, 9539, 10429, 15434, 16874, 24973, 27303, 40407, 44177, 65380, 71480, 105787, 115657, 171167, 187137, 276954, 302794, 448121, 489931

Offset: 1

Mark Dols, Aug 18 2010

Generalization of A115339.

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

Cf. A115339, A000045, A000032.

LinearRecurrence[{0,1,0,1},{101,10,10,101},50] (* Paolo Xausa, Jan 04 2024 *)

Vec(-x*(91*x^3-91*x^2+10*x+101)/(x^4+x^2-1) + O(x^100)) \\ Colin Barker, Oct 03 2015

a(n) = if(n==1||n==4, 101, if(n==2||n==3, 10, a(n-2)+a(n-4))); \\ Altug Alkan, Oct 03 2015

G.f.: -x*(91*x^3-91*x^2+10*x+101) / (x^4+x^2-1). Colin Barker, Oct 03 2015

A236144 a(n) = F(floor( (n+3)/2 )) * L(floor( (n+2)/2 )) where F=Fibonacci and L=Lucas numbers.

Original entry on oeis.org

2, 2, 1, 2, 6, 9, 12, 20, 35, 56, 88, 143, 234, 378, 609, 986, 1598, 2585, 4180, 6764, 10947, 17712, 28656, 46367, 75026, 121394, 196417, 317810, 514230, 832041, 1346268, 2178308, 3524579, 5702888, 9227464, 14930351, 24157818, 39088170, 63245985, 102334154

Offset: 0

Michael Somos, Jan 19 2014

			G.f. = 2 + 2*x + x^2 + 2*x^3 + 6*x^4 + 9*x^5 + 12*x^6 + 20*x^7 + 35*x^8 + ...

G. C. Greubel, Table of n, a(n) for n = 0..2500
Index entries for linear recurrences with constant coefficients, signature (1,0,1,1).

Cf. A000032, A000045, A115008, A115339, A128534, A128535.

m:=60; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(2-x^2-x^3)/(1-x-x^3-x^4)); // G. C. Greubel, Aug 07 2018

a[ n_] := Fibonacci[ Quotient[ n + 3, 2]] LucasL[ Quotient[ n, 2]];
CoefficientList[Series[(2-x^2-x^3)/(1-x-x^3-x^4), {x, 0, 60}], x] (* G. C. Greubel, Aug 07 2018 *)

{a(n) = fibonacci( (n+3)\2 ) * (fibonacci( n\2+1 ) + fibonacci( n\2-1 ))};

x='x+O('x^60); Vec((2-x^2-x^3)/(1-x-x^3-x^4)) \\ G. C. Greubel, Aug 07 2018

G.f.: (2 - x^2 - x^3) / (1 - x - x^3 - x^4) = (1 - x) * (2 + 2*x + x^2) / ((1 + x^2) * (1 - x - x^2)).
a(n) = a(n-1) + a(n-3) + a(n-4) for all n in Z.
0 = a(n)*a(n+2) + a(n+1)*(+a(n+2) -a(n+3)) for all n in Z.
a(n) = A115008(n+2) - A115008(n+1).
a(n) = A115339(n) * A115339(n-1).
a(2*n - 1) = F(n+1) * L(n-1) = A128535(n+1). a(2*n) = F(n+1) * L(n) = A128534(n+1).
a(n) = A000045(n+1)+A057077(n). - R. J. Mathar, Sep 24 2021

cp's OEIS Frontend

A116470 All distinct Fibonacci and Lucas numbers.

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A180236 a(n) = a(n-2)+a(n-4); a(1)=a(4)=101, a(2)=a(3)=10.

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A236144 a(n) = F(floor( (n+3)/2 )) * L(floor( (n+2)/2 )) where F=Fibonacci and L=Lucas numbers.

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