A062114 -id:A062114 - cp's OEIS frontend

A014217 a(n) = floor(phi^n), where phi = (1+sqrt(5))/2 is the golden ratio.

1, 1, 2, 4, 6, 11, 17, 29, 46, 76, 122, 199, 321, 521, 842, 1364, 2206, 3571, 5777, 9349, 15126, 24476, 39602, 64079, 103681, 167761, 271442, 439204, 710646, 1149851, 1860497, 3010349, 4870846, 7881196, 12752042, 20633239, 33385281, 54018521, 87403802

Offset: 0

Clark Kimberling

a(n) = floor(lim_{k->oo} Fibonacci(k)/Fibonacci(k-n)). - Jon Perry, Jun 10 2003
For n > 1, a(n) is the maximum element in the continued fraction for A000045(n)*phi. - Benoit Cloitre, Jun 19 2005
a(n) is also the curvature (rounded down) of the circle inscribed in the n-th kite arranged in a spiral, starting with a unit circle, as shown in the illustration in the links section. - Kival Ngaokrajang, Aug 29 2013
a(n) is the n-th Lucas number (A000032) if n is odd, and a(n) is the n-th Lucas number minus 1 if n is even. (Mario Catalani's formula below expresses this fact.) This is related to the fact that the powers of phi approach the values of the Lucas numbers, the odd powers from above and the even powers from below. - Geoffrey Caveney, Apr 18 2014
a(n) is the sum of the last summands over all Arndt compositions of n (see the Checa link). - Daniel Checa, Dec 25 2023
a(n) is the number of (saturated or unsaturated) substituted N-heterocycles in chemistry (N = nitrogen). That means the number of matchings in a cycle graph when the two maximum matchings in every cycle with an even number of vertices are indistinguishable (because the corresponding resonance structures in the molecule are equivalent). - Stefan Schuster, Mar 20 2025

Alois P. Heinz, Table of n, a(n) for n = 0..4784 (first 301 terms from T. D. Noe)
Mohammad K. Azarian, Problem 123, Missouri Journal of Mathematical Sciences, Vol. 10, No. 3, Fall 1998, p. 176. Solution published in Vol. 12, No. 1, Winter 2000, pp. 61-62.
Daniel F. Checa, Arndt Compositions: Connections with Fibonacci Numbers, Statistics, and Generalizations, 2023. p. 29.
Daniel F. Checa and José L. Ramírez, Arndt Compositions: A Generating Functions Approach, Integers (2024) Vol. 24, A35. See p. 14.
G. Harman, One hundred years of normal numbers
Kival Ngaokrajang, Illustration for n = 0..7
Stefan Schuster and Tatjana Malycheva, Enumeration of saturated and unsaturated substituted N-heterocycles, arXiv:2309.02343 [q-bio.BM], 2023.
Index entries for linear recurrences with constant coefficients, signature (1,2,-1,-1).

Cf. A000032, A000045, A001622, A020956, A022319, A052952, A057146, A062114, A128440, A153382, A169985, A169986, A203976, A226328.

First differences give A181716.

a014217 n = a014217_list !! n
a014217_list = 1 : 1 : zipWith (+)
   a000035_list (zipWith (+) a014217_list $ tail a014217_list)
-- Reinhard Zumkeller, Jan 06 2012

[Floor( ((1+Sqrt(5))/2)^n ): n in [0..100]]; // Vincenzo Librandi, Apr 16 2011

A014217 := proc(n)
    option remember;
    if n <= 3 then
        op(n+1,[1,1,2,4]) ;
    else
        procname(n-1)+2*procname(n-2)-procname(n-3)-procname(n-4) ;
    end if;
end proc: # R. J. Mathar, Jun 23 2013
#
a:= n-> (<<0|1|0|0>, <0|0|1|0>, <0|0|0|1>, <-1|-1|2|1>>^n. <<1, 1, 2, 4>>)[1, 1]:
seq(a(n), n=0..40);  # Alois P. Heinz, Oct 12 2017

Table[Floor[GoldenRatio^n], {n, 0, 36}] (* Vladimir Joseph Stephan Orlovsky, Dec 12 2008 *)
LinearRecurrence[{1, 2, -1, -1}, {1, 1, 2, 4}, 40] (* Jean-François Alcover, Nov 05 2017 *)

my(x='x+O('x^44)); Vec((1-x^2+x^3)/((1+x)*(1-x)*(1-x-x^2))) \\ Joerg Arndt, Jul 10 2023

from sympy import floor, sqrt
def A014217(n): return floor(((1+sqrt(5))/2)**n) # Chai Wah Wu, Dec 17 2021

[floor(golden_ratio^n) for n in range(37)] # Danny Rorabaugh, Apr 19 2015

a(n) = a(n-1) + 2*a(n-2) - a(n-3) - a(n-4).
a(n) = a(n-1) + a(n-2) + (1-(-1)^n)/2 = a(n-1) + a(n-2) + A000035(n).
a(n) = A000032(n) - (1 + (-1)^n)/2. - Mario Catalani (mario.catalani(AT)unito.it), Jan 17 2003
G.f.: (1-x^2+x^3)/((1+x)*(1-x)*(1-x-x^2)). - R. J. Mathar, Sep 06 2008
a(2n-1) = (Fibonacci(4n+1)-2)/Fibonacci(2n+2). - Gary Detlefs, Feb 16 2011
a(n) = floor(Fibonacci(2n+3)/Fibonacci(n+3)). - Gary Detlefs, Feb 28 2011
a(2n) = Fibonacci(2*n-1) + Fibonacci(2*n+1) - 1. - Gary Detlefs, Mar 10 2011
a(n+2*k) - a(n) = A203976(k)*A000032(n+k) if k odd, a(n+2*k) - a(n) = A203976(k)*A000045(n+k) if k even, for k > 0. - Paul Curtz, Jun 05 2013
a(n) = A052952(n) - A052952(n-2) + A052952(n-3). - R. J. Mathar, Jun 13 2013
a(n+6) - a(n-6) = 40*A000045(n), case k=6 of my formula above. - Paul Curtz, Jun 13 2013
From Paul Curtz, Jun 17 2013: (Start)
a(n-3) + a(n+3) = A153382(n).
a(n-1) + a(n+2) = A022319(n). (End)
For k > 0, a(2k) = A169985(2k)-1 and a(2k+1) = A169985(2k+1) (which is equivalent to Catalani's 2003 formula). - Danny Rorabaugh, Apr 15 2015
a(n) = ((-1)^(1+n)-1)/2 + ((1-sqrt(5))/2)^n + ((1+sqrt(5))/2)^n. - Colin Barker, Nov 05 2017
a(n) = floor(2*sinh(n*arccsch(2))). - Federico Provvedi, Feb 23 2022
E.g.f.: 2*exp(x/2)*cosh(sqrt(5)*x/2) - cosh(x). - Stefano Spezia, Jul 26 2022
a(n) = floor(Fibonacci(n)*phi) + Fibonacci(n-1) = A074331(n) + A000045(n-1) = A052952(n-1) + A000045(n-1). This is the case k=1 of the formula (also found in A128440): floor(k * phi^n) = floor(Fibonacci(n)*k*phi) + Fibonacci(n-1) * k. - Chunqing Liu, Oct 03 2023

Corrected by T. D. Noe, Nov 09 2006

Edited by N. J. A. Sloane, Aug 29 2008 at the suggestion of R. J. Mathar

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

Original entry on oeis.org

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

A069403 a(n) = 2Fibonacci(2n+1) - 1.

Original entry on oeis.org

1, 3, 9, 25, 67, 177, 465, 1219, 3193, 8361, 21891, 57313, 150049, 392835, 1028457, 2692537, 7049155, 18454929, 48315633, 126491971, 331160281, 866988873, 2269806339, 5942430145, 15557484097, 40730022147, 106632582345, 279167724889, 730870592323, 1913444052081

Offset: 0

R. H. Hardin, Mar 22 2002

Half the number of n X 3 binary arrays with a path of adjacent 1's and a path of adjacent 0's from top row to bottom row.

Indices of A017245 = 9*n + 7 = 7, 16, 25, 34, for submitted A153819 = 16, 34, 88,. A153819(n) = 9*a(n) + 7 = 18*F(2*n+1) -2; F(n) = Fibonacci = A000045, 2's = A007395. Other recurrence: a(n) = 4*a(n-1) - 4*a(n-2) + a(n-3). - Paul Curtz, Jan 02 2009

Colin Barker, Table of n, a(n) for n = 0..1000
Yurii S. Bystryk, Vitalii L. Denysenko, and Volodymyr I. Ostryk, Lune and Lens Sequences, ResearchGate preprint, 2024. See pp. 41, 56.
J. Hietarinta and C.-M. Viallet, Singularity confinement and chaos in discrete systems, Physical Review Letters 81 (1998), pp. 326-328.
Index entries for linear recurrences with constant coefficients, signature (4,-4,1).

Cf. A000045, A084707.
Cf. 1 X n A000225, 2 X n A016269, vertical path of 1 A069361-A069395, vertical paths of 0+1 A069396-A069416, vertical path of 1 not 0 A069417-A069428, no vertical paths A069429-A069447, no horizontal or vertical paths A069448-A069452.
Equals A052995 - 1.
Bisection of A001595, A062114, A066983.

List([0..30], n-> 2*Fibonacci(2*n+1)-1); # G. C. Greubel, Jul 11 2019

[2*Fibonacci(2*n+1)-1: n in [0..30]]; // Vincenzo Librandi, Apr 18 2011

a[n_]:= a[n] = 3a[n-1] - 3a[n-3] + a[n-4]; a[0] = 1; a[1] = 3; a[2] = 9; a[3] = 25; Table[ a[n], {n, 0, 30}]
Table[2*Fibonacci[2*n+1]-1, {n,0,30}] (* G. C. Greubel, Apr 22 2018 *)
LinearRecurrence[{4,-4,1},{1,3,9},30] (* Harvey P. Dale, Sep 22 2020 *)

a(n) = 2*fibonacci(2*n+1)-1 \\ Charles R Greathouse IV, Jun 11 2015

Vec((1-x+x^2)/((1-x)*(1-3*x+x^2)) + O(x^30)) \\ Colin Barker, Nov 02 2016

[2*fibonacci(2*n+1)-1 for n in (0..30)] # G. C. Greubel, Jul 11 2019

a(0) = 1, a(1) = 3, a(2) = 9, a(3) = 25; a(n) = 3*a(n-1) - 3*a(n-3) + a(n-4).
a(n) = 3*a(n-1) - a(n-2) + 1 for n>1, a(1) = 3, a(0) = 0. - Reinhard Zumkeller, May 02 2006
From R. J. Mathar, Feb 23 2009: (Start)
a(n) = 4*a(n-1) - 4*a(n-2) + a(n-3).
G.f.: (1-x+x^2)/((1-x)*(1-3*x+x^2)). (End)
a(n) = 1 + 2*Sum_{k=0..n} Fibonacci(2*k) = 1+2*A027941(n). - Gary Detlefs, Dec 07 2010
a(n) = (2^(-n)*(-5*2^n -(3-sqrt(5))^n*(-5+sqrt(5)) +(3+sqrt(5))^n*(5+sqrt(5))))/5. - Colin Barker, Nov 02 2016

Simpler definition from Vladeta Jovovic, Mar 19 2003

A138606 List first F(1) odd numbers, then first F(2) even numbers (starting from 2), then the next F(3) odd numbers, then the next F(4) even numbers, etc., where F(n) = A000045(n), the n-th Fibonacci number.

Original entry on oeis.org

1, 2, 3, 5, 4, 6, 8, 7, 9, 11, 13, 15, 10, 12, 14, 16, 18, 20, 22, 24, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77

Offset: 1

Ctibor O. Zizka, May 14 2008

The original name was "FibCon sequence". However, this sequence has only a passing resemblance to Connell-like sequences (see A001614), which are all monotone, while this sequence is a bijection of natural numbers.

Fixed points of the permutation are the terms of A062114. - Ivan Neretin, Sep 04 2017

			Let us separate the positive integers into odd (A005408) and even numbers (A005843):
1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,...
2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,...
then we get the following subsequences:
S1={1}
S2={2}
S3={3,5}
S4={4,6,8}
S5={7,9,11,13,15}
S6={10,12,14,16,18,20,22,24}
...
and concatenating them S1/S2/S3/S4/S5/... gives this sequence.

Inverse: A166013. A000035(a(n)) = A000035(A072649(n)). Cf. A138607-A138609, A138612.

o = 1; e = 2; Flatten@Table[If[OddQ[n], Range[o, (o += 2 Fibonacci[n]) - 1, 2], Range[e, (e += 2 Fibonacci[n]) - 1, 2]], {n, 9}] (* Ivan Neretin, Sep 04 2017 *)

a(n) = A166012(A072649(n)-1) + 2*(n - A000045(1+A072649(n))). - Antti Karttunen, Oct 05 2009

Edited, extended and Scheme code added by Antti Karttunen, Oct 05 2009

A131326 Row sums of A131325.

Original entry on oeis.org

1, 3, 4, 9, 13, 24, 37, 63, 100, 165, 265, 432, 697, 1131, 1828, 2961, 4789, 7752, 12541, 20295, 32836, 53133, 85969, 139104, 225073, 364179, 589252, 953433, 1542685, 2496120, 4038805, 6534927, 10573732, 17108661, 27682393, 44791056, 72473449, 117264507

Offset: 0

Gary W. Adamson, Jun 28 2007

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-1,-1).

Cf. A131325, A049310, A062114, A052952, A022086.

Vec((1+2*x-x^2) / ((x-1)*(1+x)*(x^2+x-1)) + O(x^50)) \\ Colin Barker, Jul 12 2017

a(n) = A062114(n+1)+A052952(n-1).
G.f. ( 1+2*x-x^2 ) / ( (x-1)*(1+x)*(x^2+x-1) ).  - R. J. Mathar, Aug 12 2012
a(n) = 3*Fibonacci(n+1)-2 if n even, a(n) = 3*Fibonacci(n+1) if n odd. - R. J. Mathar, Aug 12 2012
From Colin Barker, Jul 12 2017: (Start)
a(n) = (3*2^(-n-1)*((1 + sqrt(5))^(n+1) - (1 - sqrt(5))^(n+1))) / sqrt(5) - 2 for n even.
a(n) = (3*2^(-n-1)*((1 + sqrt(5))^(n+1) - (1 - sqrt(5))^(n+1))) / sqrt(5) for n odd.
a(n) = a(n-1) + 2*a(n-2) - a(n-3) - a(n-4) for n>3.
(End)

A131324 2*A049310 - A000012(signed).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 5, 1, 1, 1, 5, 1, 7, 1, 1, 1, 1, 11, 1, 9, 1, 1, 1, 7, 1, 19, 1, 11, 1, 1, 1, 1, 19, 1, 29, 1, 13, 1, 1, 1, 9, 1, 39, 1, 41, 1, 15, 1, 1

Offset: 0

Gary W. Adamson, Jun 28 2007

Row sums = A062114: (1, 2, 3, 6, 9, 16, 25, ...).

			First few rows of the triangle:
  1;
  1,  1;
  1,  1,  1;
  1,  3,  1,  1;
  1,  1,  5,  1,  1;
  1,  5,  1,  7,  1,  1,
  1,  1, 11,  1,  9,  1,  1;
  1,  7,  1, 19,  1, 11,  1,  1;
  ...

Cf. A049310, A062114, A131325, A131326, A131327.

2*A049310 - A000012(signed + - + - ... by columns).

A131328 Row sums of triangle A131327.

Original entry on oeis.org

1, 4, 5, 12, 17, 32, 49, 84, 133, 220, 353, 576, 929, 1508, 2437, 3948, 6385, 10336, 16721, 27060, 43781, 70844, 114625, 185472, 300097, 485572, 785669, 1271244, 2056913, 3328160, 5385073, 8713236, 14098309, 22811548, 36909857, 59721408, 96631265, 156352676

Offset: 0

Gary W. Adamson, Jun 28 2007

a(n)/a(n-1) tends to phi. (Cf. A062114).

			a(3) = 12 = sum of row 3 terms of A131327: (3 + 5 + 3 + 1).
a(3) = (9 + 3) since we add terms of A131326: (1, 3, 4, 9, 13,...) to A052952: (0, 1, 1, 3, 4,...), getting (9 + 3 ) = 12.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-1,-1).

Cf. A062114, A052952, A131324, A131325, A131326, A131327.

Vec((1 + 3*x - x^2) / ((1 - x)*(1 + x)*(1 - x - x^2)) + O(x^50)) \\ Colin Barker, Jul 12 2017

a(n+1) = A131326(n) + A052952(n+1).
a(n) = -3*(1+(-1)^n)/2 +4*A000045(n+1). - R. J. Mathar, Aug 13 2012
G.f.: ( 1+3*x-x^2 ) / ( (x-1)*(1+x)*(x^2+x-1) ). - R. J. Mathar, Aug 13 2012
From Colin Barker, Jul 12 2017: (Start)
a(n) = (2^(1-n)*((1+sqrt(5))^(n+1) - (1-sqrt(5))^(n+1))) / sqrt(5) - 3 for n even.
a(n) = (2^(1-n)*((1+sqrt(5))^(n+1) - (1-sqrt(5))^(n+1))) / sqrt(5) for n odd.
a(n) = a(n-1) + 2*a(n-2) - a(n-3) - a(n-4) for n>3.
(End)

More terms from Colin Barker, Jul 12 2017

A134513 Triangle read by rows: T(n, k) = binomial(ceiling((n+k)/2), floor((n-k)/2)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 3, 3, 1, 1, 3, 3, 4, 4, 1, 1, 1, 6, 6, 5, 5, 1, 1, 4, 4, 10, 10, 6, 6, 1, 1, 1, 10, 10, 15, 15, 7, 7, 1, 1, 5, 5, 20, 20, 21, 21, 8, 8, 1, 1, 1, 15, 15, 35, 35, 28, 28, 9, 9, 1, 1, 6, 6, 35, 35, 56, 56, 36, 36, 10, 10, 1, 1

Offset: 0

Gary W. Adamson, Oct 28 2007

Old name: abs(A049310 * A097806).
Equivalently, T(n,k) = A168561(n,k) + A168561(n,k+1).
Row sums = A062114: (1, 2, 3, 6, 9, 16, 25, 42, 67, ...).
Triangle A046854 = abs(A097806 * A049310).

			First few rows of the triangle:
  1;
  1,  1;
  1,  1,  1;
  2,  2,  1,  1;
  1,  3,  3,  1,  1;
  3,  3,  4,  4,  1,  1;
  1,  6,  6,  5,  5,  1,  1;
  4,  4, 10, 10,  6,  6,  1,  1;
  1, 10, 10, 15, 15,  7,  7,  1,  1;
  ...

Cf. A046854, A049310, A062114, A097806.

abs(A049310 * A097806) as infinite lower triangular matrices.

Better definition, offset changed to 0, and more terms from Jinyuan Wang, Jan 25 2025

A166013 Inverse permutation to A138606.

Original entry on oeis.org

1, 2, 3, 5, 4, 6, 8, 7, 9, 13, 10, 14, 11, 15, 12, 16, 21, 17, 22, 18, 23, 19, 24, 20, 25, 34, 26, 35, 27, 36, 28, 37, 29, 38, 30, 39, 31, 40, 32, 41, 33, 42, 55, 43, 56, 44, 57, 45, 58, 46, 59, 47, 60, 48, 61, 49, 62, 50, 63, 51, 64, 52, 65, 53, 66, 54, 67, 89, 68, 90, 69, 91

Offset: 1

Antti Karttunen, Oct 05 2009

nonn

Fixed points of the permutation are the terms of A062114. - Ivan Neretin, Sep 04 2017

Inverse: A138606.

A279890 Expansion of x(1 - x + 2x^3 - x^4)/((1 - x)(1 + x)(1 - x + x^2)*(1 - x - x^2)).

Original entry on oeis.org

0, 1, 1, 2, 4, 7, 12, 19, 31, 50, 82, 133, 216, 349, 565, 914, 1480, 2395, 3876, 6271, 10147, 16418, 26566, 42985, 69552, 112537, 182089, 294626, 476716, 771343, 1248060, 2019403, 3267463, 5286866, 8554330, 13841197, 22395528, 36236725, 58632253, 94868978, 153501232, 248370211, 401871444, 650241655, 1052113099, 1702354754

Offset: 0

Ilya Gutkovskiy, Dec 22 2016

The integer part of the harmonic mean of Fibonacci(n), Fibonacci(n+1) and Fibonacci(n+2).
The o.g.f. for the numerators of the fractional part of the harmonic mean of Fibonacci(n), Fibonacci(n+1) and Fibonacci(n+2) is 6*x/((1 + x - x^2)*(1 - 4*x - x^2)).
The o.g.f. for the denominators of the fractional part of the harmonic mean of Fibonacci(n), Fibonacci(n+1) and Fibonacci(n+2) is (1 + 3*x - x^2)/((1 + x)*(1 - 3*x + x^2)).
Convolution of Fibonacci numbers and periodic sequence [1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, ...].

			a(1) = floor(3/(1/F(1)+1/F(2)+1/F(3))) = floor(3/(1/1+1/1+1/2)) = 1;
a(2) = floor(3/(1/F(2)+1/F(3)+1/F(4))) = floor(3/(1/1+1/2+1/3)) = 1;
a(3) = floor(3/(1/F(3)+1/F(4)+1/F(5))) = floor(3/(1/2+1/3+1/5)) = 2, etc.

Eric Weisstein's World of Mathematics, Fibonacci Number
Eric Weisstein's World of Mathematics, Harmonic Mean
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,2,0,-1).

Cf. A000045, A001622, A004695, A065563, A236428.
Cf. A062114 (the integer part of the harmonic mean of Fibonacci(n+1) and Fibonacci(n+2) for n>0).
Cf. A074331 (the integer part of the geometric mean of Fibonacci(n), Fibonacci(n+1) and Fibonacci(n+2)).

LinearRecurrence[{2, 0, -2, 2, 0, -1}, {0, 1, 1, 2, 4, 7}, 46]
Table[Floor[3 Fibonacci[n] Fibonacci[n + 1] Fibonacci[n + 2]/(2 Fibonacci[n + 1] Fibonacci[n + 2] - (-1)^n)], {n, 0, 45}]

concat(0, Vec((x*(1-x+2*x^3-x^4)/((1-x)*(1+x)*(1-x+x^2))) + O(x^40))) \\ Felix Fröhlich, Dec 22 2016

G.f.: x*(1 - x + 2*x^3 - x^4)/((1 - x)*(1 + x)*(1 - x + x^2)*(1 - x - x^2)).
a(n) = 2*a(n-1) - 2*a(n-3) + 2*a(n-4) - a(n-6).
a(n) = (9*sqrt(5)*(((1 + sqrt(5))/2)^n - ((1 - sqrt(5))/2)^n) + 5*((-1)^n + 2*cos(Pi*n/3) - 3))/30.
a(n) = floor(3*F(n)*F(n+1)*F(n+2)/(2*F(n+1)*F(n+2)-(-1)^n)), where F(n) is the n-th Fibonacci number (A000045).
a(n) = floor(3*A065563(n)/A236428(n+1)).
a(n) = 3*A000045(n)/2 + ((-1)^n + 2*cos(Pi*n/3) - 3)/6.
a(n) ~ 3*phi^n/(2*sqrt(5)), where phi is the golden ratio (A001622).
Lim_{n->infinity} a(n+1)/a(n) = phi.

cp's OEIS Frontend

A014217 a(n) = floor(phi^n), where phi = (1+sqrt(5))/2 is the golden ratio.

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

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A069403 a(n) = 2*Fibonacci(2*n+1) - 1.

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A138606 List first F(1) odd numbers, then first F(2) even numbers (starting from 2), then the next F(3) odd numbers, then the next F(4) even numbers, etc., where F(n) = A000045(n), the n-th Fibonacci number.

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A131326 Row sums of A131325.

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A131324 2*A049310 - A000012(signed).

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A069403 a(n) = 2Fibonacci(2n+1) - 1.

A279890 Expansion of x(1 - x + 2x^3 - x^4)/((1 - x)(1 + x)(1 - x + x^2)*(1 - x - x^2)).