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A002203 Companion Pell numbers: a(n) = 2*a(n-1) + a(n-2), a(0) = a(1) = 2.
(Formerly M0360 N0136)
159
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
Also named "Pell-Lucas numbers", apparently by Hoggatt and Alexanderson (1976), after the English mathematician John Pell (1611-1685) and the French mathematician Édouard Lucas (1842-1891). - Amiram Eldar, Oct 02 2023
REFERENCES
Paul Bachmann, Niedere Zahlentheorie (1902, 1910), reprinted Chelsea, NY, 1968, vol. 2, p. 76.
Paulo 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
Gerald L. Alexanderson, Problem B-102, Fib. Quart., 4 (1966), 373.
Dorin Andrica and Ovidiu Bagdasar, Recurrent Sequences: Key Results, Applications, and Problems, Springer (2020), p. 39.
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.
Hacène Belbachir, Amine Belkhir, and Ihab-Eddin Djellas, Permanent of Toeplitz-Hessenberg Matrices with Generalized Fibonacci and Lucas entries, Applications and Applied Mathematics: An International Journal (AAM 2022), Vol. 17, Iss. 2, Art. 15, 558-570.
Pooja Bhadouria, Deepika Jhala, and Bijendra 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}.
Abdullah Çağman, Repdigits as sums of three Half-companion Pell numbers, Miskolc Mathematical Notes (Hungary, 2023) Vol. 24, No. 2, 687-697, MMN-4143.
Kwang-Wu Chen and Yu-Ren Pan, Greatest Common Divisors of Shifted Horadam Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.5.8.
Robert Dougherty-Bliss, Experimental Methods in Number Theory and Combinatorics, Ph. D. Dissertation, Rutgers Univ. (2024). See p. 38.
Sergio 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. [Wayback Machine link]
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. Cetin Firengiz and 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.
Taras Goy and Mark Shattuck, Determinants of Toeplitz-Hessenberg Matrices with Generalized Leonardo Number Entries, Ann. Math. Silesianae (2023). See p. 3.
Verner E. Hoggatt, Jr., and Gerald L. Alexanderson, Sums of Partition Sets in Generalized Pascal Triangles I, The Fibonacci Quarterly, Vol. 14, No. 2 (1976), pp. 117-125.
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, Amer. J. Math., 1 (1878), 184-240.
Edouard Lucas, Théorie des Fonctions Numériques Simplement Périodiques, I", Amer. J. Math., 1 (1878), 184-240 and 289-321.
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.
aBa Mbirika, Janeè Schrader, and Jürgen Spilker, Pell and associated Pell braid sequences as GCDs of sums of k consecutive Pell, balancing, and related numbers, arXiv:2301.05758 [math.NT], 2023. See also J. Int. Seq. (2023) Vol. 26, Art. 23.6.4.
Ezgi Kantarcı Oguz, Cem Yalım Özel, and Mohan Ravichandran, Chainlink Polytopes, Séminaire Lotharingien de Combinatoire, Proc. 35th Conf. Formal Power Series and Alg. Comb. (Davis, 2023) Vol. 89B, Art. #67.
Hideyuki Ohtsuka, Problem 12090, The American Mathematical Monthly, Vol. 126, No. 2 (2019), p. 180; A Pell-Lucas Computation of Pi, Solution to Problem 12090 by M. Vowe, ibid., Vol. 127, No. 7 (2020), pp. 666-667.
Neşe Ömür, Gökhan Soydan, Yücel Türker Ulutaş, and Yusuf Doğru, On triangles with coordinates of vertices from the terms of the sequences {U_kn} and {V_kn}, Matematičke Znanosti, Vol. 24 = 542(2020), 15-27.
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; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Mihai Prunescu and Lorenzo Sauras-Altuzarra, On the representation of C-recursive integer sequences by arithmetic terms, arXiv:2405.04083 [math.LO], 2024. See p. 15.
Salah Eddine Rihane and Alain Togbé, On the intersection of Padovan, Perrin sequences and Pell, Pell-Lucas sequences, Annales Mathematicae et Informaticae (2021).
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, On Generalized (r, s)-numbers, Int. J. Adv. Appl. Math. and Mech. (2020) Vol. 8, No. 1, 1-14.
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.
Anetta Szynal-Liana, Iwona Włoch, and Mirosław Liana, Generalized commutative quaternion polynomials of the Fibonacci type, Annales Math. Sect. A, Univ. Mariae Curie-Skłodowska (Poland 2022) Vol. 76, No. 2, 33-44.
Ahmet Tekcan, Merve Tayat, and Meltem E. Özbek, 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.
Zongzhen Xie, Hanpeng Gao, and Zhaoyong Huang, Tilting modules over Auslander algebras of Nakayama algebras with radical cube zero, Nanjing University (China, 2020).
Fatih Yılmaz and Mustafa Özkan, On the Generalized Gaussian Fibonacci Numbers and Horadam Hybrid Numbers: A Unified Approach, Axioms (2022) Vol. 11, No. 6, 255.
Abdelmoumène Zekiri, Farid Bencherif, and Rachid Boumahdi, Generalization of an Identity of Apostol, J. Int. Seq., Vol. 21 (2018), Article 18.5.1.
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), 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
a(n) = (-1)^n * (a(n)^3 - a(3*n))/3. - Greg Dresden, Jun 16 2021
a(n) = (a(n+2) + a(n-2))/6 for n >= 2. - Greg Dresden, Jun 23 2021
From Greg Dresden and Tongjia Rao, Sep 09 2021: (Start)
a(3n+2)/a(3n-1) = [14, ..., 14, -3] with (n+1) 14's.
a(3n+3)/a( 3n ) = [14, ..., 14, 7] with n 14's.
a(3n+4)/a(3n+1) = [14, ..., 14, 17] with n 14's. (End)
From Peter Bala, Nov 16 2022: (Start)
a(n) = trace([2, 1; 1, 0]^n) for n >= 1.
The Gauss congruences hold: a(n*p^k) == a(n*p^(k-1)) (mod p^k) for all positive integers n and k and all primes p.
a(3^n) == A271222(n) (mod 3^n). (End)
Sum_{n>=1} arctan(2/a(n))*arctan(2/a(n+1)) = Pi^2/32 (A244854) (Ohtsuka, 2019). - Amiram Eldar, Feb 11 2024
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
a := n -> 2*I^n*ChebyshevT(n, -I):
seq(simplify(a(n)), n = 0..30); # Peter Luschny, Dec 03 2023
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
(PARI) a(n) = my(w=quadgen(8)); (1+w)^n + (1-w)^n; \\ Michel Marcus, Jun 17 2021
CROSSREFS
Equals twice A001333.
Bisections are A003499 and A077444.
Sequence in context: A071109 A005310 A248096 * A300863 A278331 A097341
KEYWORD
nonn,easy,changed
AUTHOR
EXTENSIONS
More terms from Larry Reeves (larryr(AT)acm.org), Dec 03 2001
STATUS
approved

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Last modified June 24 06:03 EDT 2024. Contains 373661 sequences. (Running on oeis4.)