A168309 -id:A168309 - cp's OEIS frontend

A052952 a(n) = Fibonacci(n+2) - (1-(-1)^n)/2.

1, 1, 3, 4, 8, 12, 21, 33, 55, 88, 144, 232, 377, 609, 987, 1596, 2584, 4180, 6765, 10945, 17711, 28656, 46368, 75024, 121393, 196417, 317811, 514228, 832040, 1346268, 2178309, 3524577, 5702887, 9227464, 14930352, 24157816, 39088169, 63245985, 102334155

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Equals row sums of triangle A173284. - Gary W. Adamson, Feb 14 2010
The Kn21 sums (see A180662 for definition) of the 'Races with Ties' triangle A035317 produce this sequence. - Johannes W. Meijer, Jul 20 2011
a(n-1), for n >= 1, gives the number of compositions of n with relative prime parts, and parts not exceeding 2. See the row sums of triangle A030528 where for even n the leading 1 is missing. - Wolfdieter Lang, Jul 27 2023

			G.f. = 1 + x + 3*x^2 + 4*x^3 + 8*x^4 + 12*x^5 + 21*x^6 + 33*x^7 + ...

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1023
K. Kuhapatanakul, On the Sums of Reciprocal Generalized Fibonacci Numbers, J. Int. Seq. 16 (2013) #13.7.1, eq (1).
Steven Linton, James Propp, Tom Roby, and Julian West, Equivalence Classes of Permutations under Various Relations Generated by Constrained Transpositions, Journal of Integer Sequences, Vol. 15 (2012), #12.9.1.
H. Ohtsuka and S. Nakamura, On the sum of reciprocal sums of Fibonacci numbers, Fibonacci Quart. 46/47 (2008/2009), 153-159.
Index entries for linear recurrences with constant coefficients, signature (1,2,-1,-1).

Cf. A062114, A173284, A059841, A014217, A180662, A035317.
Partial sums of A008346, first differences of A129696.
Cf. also A000032, A000045, A030528.
Cf. A001595, A054450, A066983, A074331, A168309.

List([0..40], n-> Fibonacci(n+2) -(1-(-1)^n)/2); # G. C. Greubel, Jul 10 2019

a052952 n = a052952_list !! n
a052952_list = 1 : 1 : zipWith (+)
   a059841_list (zipWith (+) a052952_list $ tail a052952_list)
-- Reinhard Zumkeller, Jan 06 2012

[Fibonacci(n+2)-(1-(-1)^n)/2: n in [0..40]]; // Vincenzo Librandi, Dec 02 2016

A052952 :=proc(n)
    option remember;
    local t1;
    if n <= 1 then
        return 1 ;
    fi:
    if n mod 2 = 1 then
        t1:=0
    else
        t1:=1;
    fi:
    procname(n-1)+procname(n-2)+t1;
end proc;
seq(A052952(n), n=0..40) ; # N. J. A. Sloane, May 25 2008

Table[Fibonacci[n+2] -(1-(-1)^n)/2, {n, 0, 40}] (* Vincenzo Librandi, Dec 02 2016 *)
Sum[(-1)^k*Fibonacci[Range[2,41], 1-k], {k,0,1}] (* G. C. Greubel, Oct 21 2019 *)
CoefficientList[Series[1/((1-x-x^2)*(1-x^2)),{x,0,40}],x] (* Harvey P. Dale, Sep 12 2020 *)

PARI
```
{a(n) = fibonacci(n+2) - n%2};
```

[fibonacci(n+2) -(1-(-1)^n)/2 for n in (0..40)] # G. C. Greubel, Jul 10 2019

G.f.: 1/((1-x-x^2)*(1-x^2)).
a(n) = A074331(n+1).
a(n) = A054450(n+1, 1) (second column of triangle).
a(n) = 2*a(n-2) + a(n-3) + 1, with a(0)=1, a(1)=1, a(2)=3.
a(n) = Sum_{alpha=RootOf(-1+z+z^2)} (3+alpha)*alpha^(-1-n)/3 - Sum_{beta=RootOf(-1+z^2)} beta^(-1-n)/2.
a(2*k) = Sum_{j=0..k} F(2*j+1) = F(2*(k+1)) for k >= 0; a(2*k-1) = Sum_{j=0..k} F(2*j) = F(2*k+1)-1 for k >= 1 (F = A000045, Fibonacci numbers).
a(n) = a(n-1) + a(n-2) + (1+(-1)^n)/2.
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k+1, k). - Paul Barry, Oct 23 2004
a(n) = floor(phi^(n+2) / sqrt(5)), where phi is the golden ratio: phi = (1+sqrt(5))/2. - Reinhard Zumkeller, Apr 19 2005
a(n) = Fibonacci(n+1) + a(n-2) with n>1, a(0)=a(1)=1. - Zerinvary Lajos, Mar 17 2008
a(n) = floor(Fibonacci(n+3)^2/Fibonacci(n+4)). - Gary Detlefs, Nov 29 2010
a(n) = (A001595(n+3) - A066983(n+4))/2. - Gary Detlefs, Dec 19 2010
a(4*n) = F(4*n+2); a(4*n+1) = F(4*n+3) - 1; a(4*n+2) = F(4*n+4); a(4*n+3) = F(4*n+5) - 1. - Johannes W. Meijer, Jul 20 2011
a(n+1) = a(n) + a(n-1) + A059841(n+1). - Reinhard Zumkeller, Jan 06 2012
a(n) = floor(|F((1+i)*(n+2))|), n >= 0, with the complex Fibonacci function F: C -> C, z -> F(z) with F(z) := (exp(log(phi)*z) - exp(i*Pi*z)*exp(-log(phi)*z))/(2*phi-1) with the modulus |z|, the imaginary unit i and the golden section phi:=(1+sqrt(5))/2. A Conjecture: For F(z) see, e.g., the T. Koshy reference. ch. 45, p. 523, where F is called f, given in A000045. - Wolfdieter Lang, Jul 24 2012
5*a(n) = (L(n+3)-1)*(L(n+4)+3) -14 -Sum_{k=0..n} L(k+1)*L(k+5) = (L(n+3)-1)*(L(n+4)+3) -L(2*n+7) +A168309(n), where L=A000032. - J. M. Bergot, Jun 13 2014
a(n) = floor(phi*Fibonacci(n+1)), where phi is the golden section. - Michel Dekking, Dec 02 2016
a(n) = -(-1)^n * a(-4-n) for all n in Z. - Michael Somos, Dec 03 2016
a(n) = Sum_{k=0..n} Sum_{i=0..n} C(n-k-1,k-i). - Wesley Ivan Hurt, Sep 21 2017
a(n) = floor(1/(Sum_{k>=n+4} 1/Fibonacci(k))) [Ohtsuka and Nakamura]. - Michel Marcus, Aug 09 2018
a(n) = floor(abs(chebyshevU(n/2, 3/2))). - Federico Provvedi, Feb 23 2022
E.g.f.: exp(x/2)*(5*cosh(sqrt(5)*x/2) + 3*sqrt(5)*sinh(sqrt(5)*x/2))/5 - sinh(x). - Stefano Spezia, Mar 09 2024

Additional formulas and more terms from Wolfdieter Lang, May 02 2000

Better description from Olivier Gérard, Jun 05 2001

A168230 a(n) = n + 2 - a(n-1) for n>1; a(1) = 0.

Original entry on oeis.org

0, 4, 1, 5, 2, 6, 3, 7, 4, 8, 5, 9, 6, 10, 7, 11, 8, 12, 9, 13, 10, 14, 11, 15, 12, 16, 13, 17, 14, 18, 15, 19, 16, 20, 17, 21, 18, 22, 19, 23, 20, 24, 21, 25, 22, 26, 23, 27, 24, 28, 25, 29, 26, 30, 27, 31, 28, 32, 29, 33, 30, 34, 31, 35, 32, 36, 33, 37, 34, 38, 35, 39, 36, 40, 37

Offset: 1

Vincenzo Librandi, Nov 21 2009

Interleaving of A001477 and A000027 without first three terms.
Binomial transform of 0, 4 followed by a signed version of A005009.
Inverse binomial transform of A034007 without first and third term.

			a(2) = 2+2-a(1) = 4-0 = 4; a(3) = 3+2-a(2) = 5-4 = 1.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A001477 (nonnegative integers), A000027 (positive integers), A168309 (repeat 4,-3), A005009 (7*2^n), A034007 (first differences of A045891).

[ n eq 1 select 0 else -Self(n-1)+n+2: n in [1..75] ];

a=3; Table[a=n-a, {n, 3, 200}] (* Vladimir Joseph Stephan Orlovsky, Nov 22 2009 *)
CoefficientList[Series[x (4 - 3 x) / ((1 + x) (1 - x)^2),{x, 0, 100}], x] (* Vincenzo Librandi, Sep 16 2013 *)
LinearRecurrence[{1,1,-1}, {0, 4, 1}, 50] (* G. C. Greubel, Jul 16 2016 *)
nxt[{n_,a_}]:={n+1,n+3-a}; NestList[nxt,{1,0},80][[All,2]] (* Harvey P. Dale, May 28 2021 *)

Vec(x^2*(4-3*x)/((1+x)*(1-x)^2) + O(x^100)) \\ Colin Barker, Nov 08 2014

G.f.: x^2*(4 - 3*x)/((1+x)*(1-x)^2).
a(n) = (7*(-1)^n + 2*n + 5)/4.
a(n) = a(n-2) + 1 for n>2; a(1)=0, a(2)=4.
a(n+1) - a(n) = A168309(n).
a(n) = a(n-1) + a(n-2) - a(n-3). - Colin Barker, Nov 08 2014
E.g.f.: (1/4)*(7 - 12*exp(x) + (5 + 2*x)*exp(2*x))*exp(-x). - G. C. Greubel, Jul 16 2016
Sum_{n>=2} (-1)^(n+1)/a(n) = 11/6. - Amiram Eldar, Feb 23 2023

Edited, three comments, four formulas, MAGMA program added by Klaus Brockhaus, Nov 22 2009

A168330 Period 2: repeat [3, -2].

Original entry on oeis.org

3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2, 3, -2

Offset: 1

Klaus Brockhaus, Nov 23 2009

Interleaving of A010701 and -A007395.
Binomial transform of 3 followed by a signed version of A020714.
Inverse binomial transform of 3 followed by A000079.
A084964 without first two terms gives partial sums.

Index entries for linear recurrences with constant coefficients, signature (0,1).

Cf. A168309 (repeat 4, -3), A010701 (all 3's sequence), A007395 (all 2's sequence), A010716 (all 5's sequence), A020714 (5*2^n), A000079 (powers of 2), A084964 (follow n+2 by n).

Magma
```
&cat[[3,-2]: n in [1..42]];
```

[n eq 1 select 3 else -Self(n-1)+1:n in [1..84]];

[(-5*(-1)^n+1)/2: n in [1..100]]; // Vincenzo Librandi, Jul 19 2016

LinearRecurrence[{0, 1}, {3, -2}, 25] (* G. C. Greubel, Jul 18 2016 *)
PadRight[{},120,{3,-2}] (* Harvey P. Dale, Oct 05 2016 *)

a(n)=3-n%2*5 \\ Charles R Greathouse IV, Jul 13 2016

a(n) = (-5*(-1)^n + 1)/2.
a(n+1) - a(n) = 5*(-1)^n.
a(n) = -a(n-1) + 1 for n > 1; a(1) = 3.
a(n) = a(n-2) for n > 2; a(1) = 3, a(2) = -2.
G.f.: x*(3 - 2*x)/((1-x)*(1+x)).
a(n) = A049071(n). - R. J. Mathar, Nov 25 2009
E.g.f.: (1/2)*(1 - exp(-x))*(5 + exp(x)). - G. C. Greubel, Jul 18 2016

cp's OEIS Frontend

A052952 a(n) = Fibonacci(n+2) - (1-(-1)^n)/2.

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A168230 a(n) = n + 2 - a(n-1) for n>1; a(1) = 0.

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A168330 Period 2: repeat [3, -2].

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Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Magma

Magma

Mathematica

PARI

Formula