login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A002203 Companion Pell numbers: a(n) = 2*a(n-1) + a(n-2), a(0) = a(1) = 2.
(Formerly M0360 N0136)
114
2, 2, 6, 14, 34, 82, 198, 478, 1154, 2786, 6726, 16238, 39202, 94642, 228486, 551614, 1331714, 3215042, 7761798, 18738638, 45239074, 109216786, 263672646, 636562078, 1536796802, 3710155682, 8957108166, 21624372014, 52205852194, 126036076402, 304278004998 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Also the number of matchings (independent edge sets) of the n-sunlet graph. - Eric W. Weisstein, Mar 09 2016

Apart from first term, same as A099425. - Peter Shor, May 12 2005

The signed sequence 2, -2, 6, -14, 34, -82, 198, -478, 1154, -2786, ... is the Lucas V(-2,-1) sequence. - R. J. Mathar, Jan 08 2013

REFERENCES

P. Bachmann, Niedere Zahlentheorie (1902, 1910), reprinted Chelsea, NY, 1968, vol. 2, p. 76.

E. Lucas, "Théorie des Fonctions Numériques Simplement Périodiques, I", Amer. J. Math., 1 (1878), 184-240. Translated as E. Lucas, The Theory of Simply Periodic Numerical Functions, Fibonacci Association, 1969.

P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 43.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

G. L. Alexanderson, Problem B-102, Fib. Quart., 4 (1966), 373.

Jeremiah Bartz, Bruce Dearden, and Joel Iiams, Counting families of generalized balancing numbers, The Australasian Journal of Combinatorics (2020) Vol. 77, Part 3, 318-325.

P. Bhadouria, D. Jhala, and B. Singh, Binomial Transforms of the k-Lucas Sequences and its Properties, The Journal of Mathematics and Computer Science (JMCS), Volume 8, Issue 1, Pages 81-92; sequence L_{2,n}.

Kwang-Wu Chen, Yu-Ren Pan, Greatest Common Divisors of Shifted Horadam Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.5.8.

S. Falcon, On The Generating Functions of the Powers of the K-Fibonacci Numbers, Scholars Journal of Engineering and Technology (SJET), 2014; 2 (4C):669-675.

Bakir Farhi, Summation of Certain Infinite Lucas-Related Series, J. Int. Seq., Vol. 22 (2019), Article 19.1.6.

Bernadette Faye, and Florian Luca, Pell Numbers whose Euler Function is a Pell Number, arXiv:1508.05714 [math.NT], 2015.

M. C. Firengiz, A. Dil, Generalized Euler-Seidel method for second order recurrence relations, Notes on Number Theory and Discrete Mathematics, Vol. 20, 2014, No. 4, 21-32.

Refik Keskin and Olcay Karaatli, Some New Properties of Balancing Numbers and Square Triangular Numbers, Journal of Integer Sequences, Vol. 15 (2012), Article #12.1.4.

Tanya Khovanova, Recursive Sequences

Edouard Lucas, Théorie des Fonctions Numériques Simplement Périodiques, I", Amer. J. Math., 1 (1878), 184-240.

Edouard Lucas, The Theory of Simply Periodic Numerical Functions, Fibonacci Association, 1969. English translation of article "Théorie des Fonctions Numériques Simplement Périodiques, I", Amer. J. Math., 1 (1878), 184-240.

Arzu Özkoç, Some algebraic identities on quadra Fibona-Pell integer sequence, Advances in Difference Equations, 2015, 2015:148.

Serge Perrine, About the diophantine equation z^2 = 32y^2 - 16, SCIREA Journal of Mathematics (2019) Vol. 4, Issue 5, 126-139.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

Yüksel Soykan, On Summing Formulas For Generalized Fibonacci and Gaussian Generalized Fibonacci Numbers, Advances in Research (2019) Vol. 20, No. 2, 1-15, Article AIR.51824.

Yüksal Soykan, On Summing Formulas for Horadam Numbers, Asian Journal of Advanced Research and Reports (2020) Vol. 8, Issue 1, 45-61.

Yüksel Soykan, Generalized Fibonacci Numbers: Sum Formulas, Journal of Advances in Mathematics and Computer Science (2020) Vol. 35, No. 1, 89-104.

Yüksel Soykan, Closed Formulas for the Sums of Squares of Generalized Fibonacci Numbers, Asian Journal of Advanced Research and Reports (2020) Vol. 9, No. 1, 23-39, Article no. AJARR.55441.

Yüksel Soykan, A Study on Generalized Fibonacci Numbers: Sum Formulas Sum_{k=0..n} k * x^k * W_k^3 and Sum_{k=1..n} k * x^k W_-k^3 for the Cubes of Terms, Earthline Journal of Mathematical Sciences (2020) Vol. 4, No. 2, 297-331.

Yüksel Soykan, Mehmet Gümüş, and Melih Göcen, A Study On Dual Hyperbolic Generalized Pell Numbers, Zonguldak Bülent Ecevit University (Zonguldak, Turkey, 2019).

Robin James Spivey, Close encounters of the golden and silver ratios, Notes on Number Theory and Discrete Mathematics (2019) Vol. 25, No. 3, 170-184.

A. Tekcan, M. Tayat, and M. E. Ozbek, The diophantine equation 8x^2-y^2+8x(1+t)+(2t+1)^2=0 and t-balancing numbers, ISRN Combinatorics, Volume 2014, Article ID 897834, 5 pages.

Eric Weisstein's World of Mathematics, Independent Edge Set

Eric Weisstein's World of Mathematics, Matching

Eric Weisstein's World of Mathematics, Pell Number

Eric Weisstein's World of Mathematics, Sunlet Graph

Wikipedia, Lucas sequence

Abdelmoumène Zekiri, Farid Bencherif, Rachid Boumahdi, Generalization of an Identity of Apostol, J. Int. Seq., Vol. 21 (2018), Article 18.5.1.

Index entries for Lucas sequences

Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)

Index entries for linear recurrences with constant coefficients, signature (2,1).

FORMULA

a(n) = 2 * A001333(n).

a(n) = A100227(n) + 1.

O.g.f.: (2 - 2*x)/(1 - 2*x - x^2). - Simon Plouffe in his 1992 dissertation

a(n) = (1 + sqrt(2))^n + (1 - sqrt(2))^n. - Mario Catalani (mario.catalani(AT)unito.it), Mar 17 2003

a(n) = A000129(2*n)/A000129(n), n > 0. - Paul Barry, Feb 06 2004

From Miklos Kristof, Mar 19 2007: (Start)

Given F(n) = A000129(n), the Pell numbers, and L(n) = a(n), the companion Pell numbers, then:

L(n+m) + (-1)^m*L(n-m) = L(n)*L(m).

L(n+m) - (-1)^m*L(n-m) = 8*F(n)*F(m).

L(n+m+k) + (-1)^k*L(n+m-k) + (-1)^m*(L(n-m+k) + (-1)^k*L(n-m-k)) = L(n)*L(m)*L(k).

L(n+m+k) - (-1)^k*L(n+m-k) + (-1)^m*(L(n-m+k) - (-1)^k*L(n-m-k)) = 8*F(n)*L(m)*F(k).

L(n+m+k) + (-1)^k*L(n+m-k) - (-1)^m*(L(n-m+k) + (-1)^k*L(n-m-k)) = 8*F(n)*F(m)*L(k).

L(n+m+k) - (-1)^k*L(n+m-k) - (-1)^m*(L(n-m+k) - (-1)^k*L(n-m-k)) = 8*L(n)*F(m)*F(k).

(End)

a(n) = 2*(A000129(n+1) - A000129(n)). - R. J. Mathar, Nov 16 2007

G.f.: G(0), where G(k) = 1 + 1/(1 - x*(2*k - 1)/(x*(2*k + 1) - 1/G(k + 1))); (continued fraction). - Sergei N. Gladkovskii, Jun 19 2013

a(n) = [x^n] ( (1 + 2*x + sqrt(1 + 4*x + 8*x^2))/2 )^n for n >= 1. - Peter Bala, Jun 23 2015

From Kai Wang, Jan 14 2020: (Start)

A000129(m - n) = ((-1)^n * (A000129(m) * a((n) - a((m) * A000129(n))/2.

A000129(m + n) = (A000129(m) * a((n) + a((m)*A000129(n))/2.

a(n)^2 - a(n + 1) * a(n - 1) = (-1)^(n) * 8.

a(n)^2 - a(n + r) * a(n - r) = (-1)^(n - r - 1) * 8 * A000129(r)^2.

a(m) * a(n + 1) - a(m + 1) * a(n) = (-1)^(n - 1) * 8 * A000129(m - n).

a(m - n) = (-1)^(n) * (a(m) * a(n) - 8 * A000129(m) * A000129(n))/2.

a(m + n) = (a(m) * a(n) + 8 * A000129(m) * A000129(n))/2.

(End)

E.g.f.: 2*exp(x)*cosh(sqrt(2)*x). - Stefano Spezia, Jan 15 2020

a(n) = A000129(n+1) + A000129(n-1) for n>0 with a(0)=2. - Rigoberto Florez, Jul 12 2020

MAPLE

A002203 := proc(n)

    option remember;

    if n <= 1 then

        2;

    else

        2*procname(n-1)+procname(n-2) ;

    end if;

end proc: # R. J. Mathar, May 11 2013

# second Maple program:

a:= n-> (<<0|1>, <1|2>>^n. <<2, 2>>)[1, 1]:

seq(a(n), n=0..30);  # Alois P. Heinz, Jan 26 2018

MATHEMATICA

Table[LucasL[n, 2], {n, 0, 30}] (* Zerinvary Lajos, Jul 09 2009 *)

LinearRecurrence[{2, 1}, {2, 2}, 50] (* Vincenzo Librandi, Aug 15 2015 *)

Table[(1 - Sqrt[2])^n + (1 + Sqrt[2])^n, {n, 0, 20}] // Expand (* Eric W. Weisstein, Oct 03 2017 *)

LucasL[Range[0, 20], 2] (* Eric W. Weisstein, Oct 03 2017 *)

CoefficientList[Series[(2 (1 - x))/(1 - 2 x - x^2), {x, 0, 20}], x] (* Eric W. Weisstein, Oct 03 2017 *)

PROG

(Sage) [lucas_number2(n, 2, -1) for n in range(0, 29)] # Zerinvary Lajos, Apr 30 2009

(Haskell)

a002203 n = a002203_list !! n

a002203_list =

   2 : 2 : zipWith (+) (map (* 2) $ tail a002203_list) a002203_list

-- Reinhard Zumkeller, Oct 03 2011

(MAGMA) I:=[2, 2]; [n le 2 select I[n] else 2*Self(n-1)+Self(n-2): n in [1..35]] // Vincenzo Librandi, Aug 15 2015

(PARI) first(m)=my(v=vector(m)); v[1]=2; v[2]=2; for(i=3, m, v[i]=2*v[i-1]+v[i-2]); v; \\ Anders Hellström, Aug 15 2015

CROSSREFS

Equals twice A001333.

Cf. A000129, A100227.

Bisections are A003499 and A077444.

Sequence in context: A071109 A005310 A248096 * A300863 A278331 A097341

Adjacent sequences:  A002200 A002201 A002202 * A002204 A002205 A002206

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Larry Reeves (larryr(AT)acm.org), Dec 03 2001

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified October 21 03:07 EDT 2020. Contains 337910 sequences. (Running on oeis4.)