A014217 -id:A014217 - cp's OEIS frontend

A005592 a(n) = F(2n+1) + F(2n-1) - 1.

1, 2, 6, 17, 46, 122, 321, 842, 2206, 5777, 15126, 39602, 103681, 271442, 710646, 1860497, 4870846, 12752042, 33385281, 87403802, 228826126, 599074577, 1568397606, 4106118242, 10749957121, 28143753122, 73681302246, 192900153617, 505019158606, 1322157322202

Offset: 0

N. J. A. Sloane

For any m, the maximum element in the continued fraction of F(2n+m)/F(m) is a(n). - Benoit Cloitre, Jan 10 2006
The continued fraction [a(n);1,a(n)-1,1,a(n)-1,...] = phi^(2n), where phi = 1.618... is the golden ratio, A001622. - Thomas Ordowski, Jun 07 2013
a(n) is the number of labeled subgraphs of the n-cycle C_n. For example, a(3)=17. There are 7 subgraphs of the triangle C_3 with 0 edges, 6 with 1 edge, 3 with 2 edges, and 1 with 3 edges (C_3 itself); here 7+6+3+1 = 17. - John P. McSorley, Oct 31 2016
a(n) equals the sum of the n-th row of triangle A277919. - John P. McSorley, Nov 25 2016

			G.f. = 1 + 2*x + 6*x^2 + 17*x^3 + 46*x^4 + 122*x^5 + 321*x^6 + 842*x^7 + ...

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (terms n = 1..200 from Vincenzo Librandi)
Yurii S. Bystryk, Vitalii L. Denysenko, and Volodymyr I. Ostryk, Lune and Lens Sequences, ResearchGate preprint, 2024. See pp. 41, 56.
M. D. McIlroy, The number of states of a dynamic storage system, Computer J., Vol. 25, No. 3 (1982), pp. 388-392.
M. D. McIlroy, The number of states of a dynamic storage system, Computer J., Vol. 25, No. 3 (1982), pp. 388-392. (Annotated scanned copy)
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992, arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Jesús Salas and Alan D. Sokal, Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models. V. Further Results for the Square-Lattice Chromatic Polynomial, J. Stat. Phys., Vol. 135 (2009), pp. 279-373; arXiv preprint, arXiv:0711.1738 [cond-mat.stat-mech], 2007-2009. Mentions this sequence. - N. J. A. Sloane, Mar 14 2014
Robert S. Seamons, Problem B-89, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 4, No. 2 (1966), p. 190; A Close Approximation, Solution to Problem B-89 by Douglas Lind, ibid., Vol. 5, No. 1 (1967), pp. 108-109.
Index entries for linear recurrences with constant coefficients, signature (4,-4,1).

Equals A004146+1 and A005248+1.
Bisection of A014217; the other bisection is A002878, which also bisects A000032.
Cf. A000045, A065034.

a005592 n = a005592_list !! (n-1)
a005592_list = map (subtract 1) $
                   tail $ zipWith (+) a001519_list $ tail a001519_list
-- Reinhard Zumkeller, Aug 09 2013

[Fibonacci(2*n+1)+Fibonacci(2*n-1)-1: n in [1..30]]; // Vincenzo Librandi, Aug 23 2011

A005592:=-(2-2*z+z**2)/(z-1)/(z**2-3*z+1); # conjectured by Simon Plouffe in his 1992 dissertation
# second Maple program:
F:= n-> (<<0|1>, <1|1>>^n)[1,2]:
a:= n-> F(2*n+1)+F(2*n-1)-1:
seq(a(n), n=0..30);  # Alois P. Heinz, Nov 04 2016

Table[Fibonacci[2n+1]+Fibonacci[2n-1]-1,{n,30}] (* Harvey P. Dale, Aug 22 2011 *)
a[n_] := LucasL[2n]-1; Array[a, 30] (* Jean-François Alcover, Dec 09 2015 *)

a(n)=fibonacci(2*n+1)+fibonacci(2*n-1)-1 \\ Charles R Greathouse IV, Aug 23 2011

[lucas_number2(n,3,1)-1 for n in range(1,29)] # Zerinvary Lajos, Jul 06 2008

a(n) = Lucas(2*n)-1, with Lucas(n)=A000032(n).
a(n) = floor(r^(2*n)), where r = golden ratio = (1+sqrt(5))/2.
a(n) = floor(Fibonacci(5*n)/Fibonacci(3*n)). - Gary Detlefs, Mar 11 2011
a(n) = +4*a(n-1) -4*a(n-2) +1*a(n-3). - Joerg Arndt, Mar 11 2011
a(n) = A001519(2*n-1) + A001519(2*n+1) - 1. - Reinhard Zumkeller, Aug 09 2013
a(n) = 3*a(n) - a(n-1) + 1; a(n) = A004146(n) + 1, n>0. - Richard R. Forberg, Sep 04 2013
a(n) = 2*cosh(2*n*arcsinh(1/2)) - 1. - Ilya Gutkovskiy, Oct 31 2016
a(n) = floor(sqrt(5)*Fibonacci(2*n)), for n > 0 (Seamons, 1966). - Amiram Eldar, Feb 05 2022

Formulae and comments by Clark Kimberling, Nov 24 2010

a(0)=1 prepended by Alois P. Heinz, Nov 04 2016

A020956 a(n) = Sum_{k>=1} floor(tau^(n-k)) where tau is A001622.

Original entry on oeis.org

1, 2, 4, 8, 14, 25, 42, 71, 117, 193, 315, 514, 835, 1356, 2198, 3562, 5768, 9339, 15116, 24465, 39591, 64067, 103669, 167748, 271429, 439190, 710632, 1149836, 1860482, 3010333, 4870830, 7881179, 12752025, 20633221, 33385263, 54018502, 87403783, 141422304

Offset: 1

Clark Kimberling

Colin Barker, Table of n, a(n) for n = 1..1000
Clark Kimberling, Problem 10520, Amer. Math. Mon. 103 (1996) p. 347.
Index entries for linear recurrences with constant coefficients, signature (2,1,-3,0,1).

Cf. A000032, A001622, A014217, A281362.

I:=[1,2,4,8,14]; [n le 5 select I[n] else 2*Self(n-1)+Self(n-2)-3*Self(n-3)+Self(n-5): n in [1..40]]; // Vincenzo Librandi, Nov 01 2016

[Lucas(n+1)-(2*n+5-(-1)^n)/4: n in [1..40]]; // G. C. Greubel, Apr 05 2024

LinearRecurrence[{2,1,-3,0,1}, {1,2,4,8,14}, 40] (* Vincenzo Librandi, Nov 01 2016 *)

Vec(x*(1-x^2+x^3)/((1-x-x^2)*(1+x)*(1-x)^2) + O(x^50)) \\ Michel Marcus, Nov 01 2016

prpr = 0
prev = 1
for n in range(2,100):
    print(prev, end=", ")
    curr = prpr+prev + n//2
    prpr = prev
    prev = curr
# Alex Ratushnyak, Jul 30 2012

[lucas_number2(n+1,1,-1) -(n+2+(n%2))//2 for n in range(1,41)] # G. C. Greubel, Apr 05 2024

G.f.: x*(1-x^2+x^3)/((1-x-x^2)*(1+x)*(1-x)^2). - Ralf Stephan, Apr 08 2004
a(n) = Lucas(n+1) - floor(n/2) - 1.
a(n) = Sum_{k=0..n-1} A014217(k).
a(n) = 2^(-2-n)*((-2)^n - 5*2^n + 2*(1-t)^(1+n) + 2*(1+t)^n + 2*t*(1+t)^n - 2^(1+n)*n) where t=sqrt(5). - Colin Barker, Feb 09 2017
From G. C. Greubel, Apr 05 2024: (Start)
a(n) = Lucas(n+1) - (1/4)*(2*n + 5 - (-1)^n).
E.g.f.: exp(x/2)*(cosh(sqrt(5)*x/2) + sqrt(5)*sinh(sqrt(5)*x/2)) - (1/2)*((x+2)*cosh(x) + (x+3)*sinh(x)). (End)

More terms from Vladeta Jovovic, Apr 04 2002

A169986 Ceiling(phi^n) where phi = (1+sqrt(5))/2.

Original entry on oeis.org

1, 2, 3, 5, 7, 12, 18, 30, 47, 77, 123, 200, 322, 522, 843, 1365, 2207, 3572, 5778, 9350, 15127, 24477, 39603, 64080, 103682, 167762, 271443, 439205, 710647, 1149852, 1860498, 3010350, 4870847, 7881197, 12752043, 20633240, 33385282

Offset: 0

N. J. A. Sloane, Sep 26 2010

Danny Rorabaugh, Table of n, a(n) for n = 0..4000
Index entries for linear recurrences with constant coefficients, signature (1,2,-1,-1).

Cf. A001622, A014217, A062724, A169985.

[1] cat [3*Fibonacci(n-1) + Fibonacci(n-2)+ n mod 2: n in [1..40]]; // Vincenzo Librandi, Apr 16 2015

Ceiling[GoldenRatio^Range[0,40]] (* or *) Join[{1},LinearRecurrence[{1,2,-1,-1},{2,3,5,7},40]] (* Harvey P. Dale, Nov 12 2014 *)

a(n)=if(n, 3*fibonacci(n-1) + fibonacci(n-2) + n%2, 1) \\ Charles R Greathouse IV, Apr 16 2015

[ceil(golden_ratio^n) for n in range(37)] # Danny Rorabaugh, Apr 16 2015

For n >= 5, a(n) = a(n-1) + 2a(n-2) - a(n-3) - a(n-4). - Charles R Greathouse IV, Oct 14 2010
a(n) = 3*Fibonacci(n-1) + Fibonacci(n-2) + (n mod 2), n>0. - Gary Detlefs, Dec 29 2010
G.f.: (-x+x^2+x^3+x^4-1) / ((1-x)*(1+x)*(x^2+x-1)). - R. J. Mathar, Jan 06 2011
a(2k) = A000032(2k) = A169985(2k) and a(2k+1) = A000032(2k+1)+1 = A169985(2k+1)+1, for k>0. - Danny Rorabaugh, Apr 15 2015

A128440 Array T(n,k) = floor(k*t^n) where t = golden ratio = (1 + sqrt(5))/2, read by descending antidiagonals.

Original entry on oeis.org

1, 3, 2, 4, 5, 4, 6, 7, 8, 6, 8, 10, 12, 13, 11, 9, 13, 16, 20, 22, 17, 11, 15, 21, 27, 33, 35, 29, 12, 18, 25, 34, 44, 53, 58, 46, 14, 20, 29, 41, 55, 71, 87, 93, 76, 16, 23, 33, 47, 66, 89, 116, 140, 152, 122, 17, 26, 38, 54, 77, 107, 145, 187, 228, 245, 199

Offset: 1

Clark Kimberling, Mar 03 2007

Row 1 = Lower Wythoff sequence = A000201; Row 2 = Upper Wythoff sequence = A001950; Column 1 = A014217 (after first term); T(n,n) = A128440(n). Every positive integer occurs exactly once in the first two rows.
Conjecture:  rows 2n-1 and 2n are disjoint for every positive integer n. - Clark Kimberling, Nov 11 2022
Stronger conjecture: for any positive integer n, if the numbers in rows 2n-1 and 2n are jointly arranged in increasing order, and each number is replaced by its position in the ordering, then the resulting two rows are identical to the first two rows. - Clark Kimberling, Nov 13 2022

			Corner:
   1    3    4    6    8    9   11   12
   2    5    7   10   13   15   18   20
   4    8   12   16   21   25   29   33
   6   13   20   27   34   41   47   54
  11   22   33   44   55   66   77   88
  17   35   53   71   89  107  125  143
  29   58   87  116  145  174  203  232
  46   93  140  187  234  281  328  375

Michel Marcus, Table of n, a(n) for n = 1..5050 (Antidiagonals n=1..100 of array, flattened).

Cf. A000045, A001622, A000201, A001950, A128439, A358359.

r = (1 + Sqrt[5])/2; t[k_, n_] := Floor[n*r^k];
Grid[Table[t[k, n], {k, 1, 10}, {n, 1, 20}]]
(* Clark Kimberling, Nov 11 2022 *)

T(n,k) = floor(k*quadgen(5)^n);
matrix(7, 7, n, k, T(n,k)) \\ Michel Marcus, Nov 14 2022

T(k,n) = k*F(n-1) + floor(k*t*F(n)), where F=A000045, the Fibonacci numbers.

A001674 a(n) = floor(sqrt( 2*Pi )^n).

Original entry on oeis.org

1, 2, 6, 15, 39, 98, 248, 621, 1558, 3906, 9792, 24546, 61528, 154230, 386597, 969056, 2429063, 6088760, 15262258, 38256809, 95895600, 240374623, 602529828, 1510318305, 3785806567, 9489609784, 23786924200, 59624976768, 149457652641, 374634777972

Offset: 0

N. J. A. Sloane

nonn

Cf. A001674 (ceiling sqrt(2 Pi)^n), A017910 (floor sqrt(2)^n), A000149 (floor e^n), A001672 (floor Pi^n), A062541 (floor (Pi*e)^n), A121831 (floor (Pi+e)^n), A032739 (floor (Pi/e)^n), A014217 (floor ((1+sqrt(5))/2)^n).

Table[Floor[Sqrt[2*Pi]^n], {n, 0, 50}] (* T. D. Noe, Aug 09 2012 *)

a(n)=(2*Pi)^(n/2)\1 \\ M. F. Hasler, May 29 2018

Edited by M. F. Hasler, May 29 2018

A062724 a(n) = floor(tau^n) + 1, where tau = (1 + sqrt(5))/2.

Original entry on oeis.org

2, 2, 3, 5, 7, 12, 18, 30, 47, 77, 123, 200, 322, 522, 843, 1365, 2207, 3572, 5778, 9350, 15127, 24477, 39603, 64080, 103682, 167762, 271443, 439205, 710647, 1149852, 1860498, 3010350, 4870847, 7881197, 12752043, 20633240, 33385282

Offset: 0

Jason Earls, Jul 15 2001

nonn

Apart from the first term, this sequence also gives the ceiling of the powers of the golden ratio (cf. A169986). - Mohammad K. Azarian, Apr 14 2008

Harry J. Smith, Table of n, a(n) for n = 0..400

Equals A014217 + 1.

Cf. A169985, A169986.

Floor[GoldenRatio^Range[0,40]]+1 (* Harvey P. Dale, Dec 18 2019 *)

j=[]; for(n=0,60,t=(1+sqrt(5))/2; j=concat(j,floor((t^n))+1)); j

{ default(realprecision, 200); t=(1 + sqrt(5))/2; p=1; for (n=0, 400, if (n, p*=t); write("b062724.txt", n, " ", p\1 + 1) ) } \\ Harry J. Smith, Aug 09 2009

a(n) = 3*Fibonacci(n-1) + Fibonacci(n-2) + (n mod 2), n > 0. - Gary Detlefs, Dec 29 2010

A122958 a(0)=1, a(n) = 2 - 2^(n-1) for n>0.

Original entry on oeis.org

1, 1, 0, -2, -6, -14, -30, -62, -126, -254, -510, -1022, -2046, -4094, -8190, -16382, -32766, -65534, -131070, -262142, -524286, -1048574, -2097150, -4194302, -8388606, -16777214, -33554430, -67108862, -134217726, -268435454, -536870910, -1073741822, -2147483646

Offset: 0

Philippe Deléham, Oct 26 2006

Take square of A014217 (1,1,2,4,6) and successive differences: a(n) is principal diagonal (k-th term of k-th row). a(n) differences: 0, -1, -2, -4, -8, -16, ... = -A131577. - Paul Curtz, Sep 26 2008

			G.f. = 1 + x - 2*x^3 - 6*x^4 - 14*x^5 - 30*x^6 - 62*x^7 - 126*x^8 - 254*x^9 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Yasemin Alp and E. Gokcen Kocer, Exponential Almost-Riordan Arrays, Results Math. (2024) Vol. 79, 173.
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Apart from signs, same as A000918.

Cf. A131577.

Join[{1}, LinearRecurrence[{3, -2}, {1, 0}, 50]] (* Vladimir Joseph Stephan Orlovsky, Feb 24 2012 *)
Join[{1},2-2^#&/@Range[0,30]] (* Harvey P. Dale, Jan 19 2021 *)

{a(n) = if( n<1, n==0, 2 - 2^(n-1))}; /* Michael Somos, Feb 08 2015 */

a(0) = 1, a(1) = 1, a(2) = 0, a(n) = 3*a(n-1) - 2*a(n-2) for n>2.
G.f.: (1 - 2*x - x^2)/(1 - 3*x + 2*x^2).
a(n) = -A000918(n-1) for n>0.
a(n+1) = 2*a(n) - 2 for n>0. - Michael Somos, Feb 08 2015
E.g.f.: exp(x)*(2 - cosh(x)). - Stefano Spezia, May 07 2023

Corrected a(22) by Vincenzo Librandi, Aug 11 2011

A152738 a(n) = floor((n^2)/phi).

Original entry on oeis.org

0, 0, 2, 5, 9, 15, 22, 30, 39, 50, 61, 74, 88, 104, 121, 139, 158, 178, 200, 223, 247, 272, 299, 326, 355, 386, 417, 450, 484, 519, 556, 593, 632, 673, 714, 757, 800, 846, 892, 940, 988, 1038, 1090, 1142, 1196, 1251, 1307, 1365, 1423, 1483, 1545, 1607, 1671

Offset: 0

Vladimir Joseph Stephan Orlovsky, Dec 12 2008

nonn

G. C. Greubel, Table of n, a(n) for n = 0..5000

Cf. A014217, A152737, A032616, A032635.

phi:=(1+Sqrt(5))/2; [Floor(n^2/phi): n in [0..30]]; // G. C. Greubel, Sep 01 2018

Mathematica
```
a[n_]:=Floor[(n^2)/GoldenRatio];
```

a(n)=n^2\((1+sqrt(5))/2) \\ Charles R Greathouse IV, Jul 29 2011

Offset corrected by B. D. Swan, Jan 03 2009

A156279 4 times the Lucas number A000032(n).

Original entry on oeis.org

8, 4, 12, 16, 28, 44, 72, 116, 188, 304, 492, 796, 1288, 2084, 3372, 5456, 8828, 14284, 23112, 37396, 60508, 97904, 158412, 256316, 414728, 671044, 1085772, 1756816, 2842588, 4599404, 7441992

Offset: 0

Paul Curtz, Feb 07 2009

This is a second kind "autosequence" whose first kind companion is A022087. - Jean-François Alcover, Aug 20 2022

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
OEIS Wiki, Autosequence
Index entries for linear recurrences with constant coefficients, signature (1,1).

Cf. A000032, A000045, A014217.

[4*Lucas(n): n in [0..30]]; // G. C. Greubel, Dec 21 2017

Table[4*LucasL[n], {n,0,30}] (* G. C. Greubel, Dec 21 2017 *)

a(n)=4*(fibonacci(n-1)+fibonacci(n+1)) \\ Charles R Greathouse IV, Oct 16 2015

a(n) = 4*A000032(n).
a(n) = a(n-1) + a(n-2).
a(n) = A014217(n+3) - A014217(n-3), with A014217(-5) = -11, A014217(-4) = 6, A014217(-3) = -4, A014217(-2) = 2, A014217(-1) = -1 extended as proposed in A153263.
G.f. 4*(-2 + x) / (-1 + x + x^2). - R. J. Mathar, Mar 11 2011
a(n) = Lucas(n+3) - Lucas(n-3), where Lucas(i) for i = 0..2 gives -4, 3, -1. - Bruno Berselli, Jul 27 2017

A181716 a(n) = a(n-1) + a(n-2) + (-1)^n, with a(0)=0 and a(1)=1.

Original entry on oeis.org

0, 1, 2, 2, 5, 6, 12, 17, 30, 46, 77, 122, 200, 321, 522, 842, 1365, 2206, 3572, 5777, 9350, 15126, 24477, 39602, 64080, 103681, 167762, 271442, 439205, 710646, 1149852, 1860497, 3010350, 4870846, 7881197, 12752042, 20633240, 33385281, 54018522, 87403802

Offset: 0

Robert G. Wilson v, Nov 07 2010

Aside from the first term, duplicate of A098600.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Aleksandar Petojević, Marjana Gorjanac Ranitović, Dragan Rastovac, and Milinko Mandić, The Golden Ratio, Factorials, and the Lambert W Function, Journal of Integer Sequences, Vol. 27 (2024), Article 24.5.7. See p. 4.
Index entries for linear recurrences with constant coefficients, signature (0,2,1).

Cf. A000032, A000045, A008346, A119282.

First differences of A014217.

I:=[0, 1, 2]; [n le 3 select I[n] else 2*Self(n-2)+Self(n-3): n in [1..40]]; // Vincenzo Librandi, Jan 09 2012

[Lucas(n-1)+(-1)^n: n in [0..40]]; // G. C. Greubel, Mar 25 2024

a[0]= 0; a[1]= 1; a[n_]:= a[n]= a[n-1] +a[n-2] +(-1)^n; Array[a,38,0]
LinearRecurrence[{0,2,1},{0,1,2},40] (* Vincenzo Librandi, Jan 09 2012 *)

[lucas_number2(n-1,1,-1)+(-1)^n for n in range(41)] # G. C. Greubel, Mar 25 2024

a(n) = a(n-1) + a(n-2) + (-1)^n.
a(n) = 2*a(n-2) + a(n-3).
a(n) - A000045(n) = A008346(n-2).
G.f.: x*(1+2*x)/(1-2*x^2-x^3). - Colin Barker, Jan 09 2012
a(n) = A000032(n-1) + (-1)^n. - G. C. Greubel, Mar 25 2024
E.g.f.: exp(x/2)*(sqrt(5)*sinh(sqrt(5)*x/2) - cosh(sqrt(5)*x/2)) + exp(-x). - Stefano Spezia, Jun 18 2024

cp's OEIS Frontend

A005592 a(n) = F(2n+1) + F(2n-1) - 1.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

Sage

Formula

Extensions

A020956 a(n) = Sum_{k>=1} floor(tau^(n-k)) where tau is A001622.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Magma

Mathematica

PARI

Python

SageMath

Formula

Extensions

A169986 Ceiling(phi^n) where phi = (1+sqrt(5))/2.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

A128440 Array T(n,k) = floor(k*t^n) where t = golden ratio = (1 + sqrt(5))/2, read by descending antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A001674 a(n) = floor(sqrt( 2*Pi )^n).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A062724 a(n) = floor(tau^n) + 1, where tau = (1 + sqrt(5))/2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula