|
%I M0360 N0136 #331 Apr 04 2024 10:54:44
%S 2,2,6,14,34,82,198,478,1154,2786,6726,16238,39202,94642,228486,
%T 551614,1331714,3215042,7761798,18738638,45239074,109216786,263672646,
%U 636562078,1536796802,3710155682,8957108166,21624372014,52205852194,126036076402,304278004998
%N Companion Pell numbers: a(n) = 2*a(n-1) + a(n-2), a(0) = a(1) = 2.
%C Also the number of matchings (independent edge sets) of the n-sunlet graph. - _Eric W. Weisstein_, Mar 09 2016
%C Apart from first term, same as A099425. - _Peter Shor_, May 12 2005
%C 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
%C 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
%D Paul Bachmann, Niedere Zahlentheorie (1902, 1910), reprinted Chelsea, NY, 1968, vol. 2, p. 76.
%D Paulo Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 43.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Reinhard Zumkeller, <a href="/A002203/b002203.txt">Table of n, a(n) for n = 0..1000</a>
%H Gerald L. Alexanderson, <a href="http://www.fq.math.ca/Scanned/4-4/elementary4-4.pdf">Problem B-102</a>, Fib. Quart., 4 (1966), 373.
%H Dorin Andrica and Ovidiu Bagdasar, <a href="https://doi.org/10.1007/978-3-030-51502-7">Recurrent Sequences: Key Results, Applications, and Problems</a>, Springer (2020), p. 39.
%H Jeremiah Bartz, Bruce Dearden, and Joel Iiams, <a href="https://ajc.maths.uq.edu.au/pdf/77/ajc_v77_p318.pdf">Counting families of generalized balancing numbers</a>, The Australasian Journal of Combinatorics (2020) Vol. 77, Part 3, 318-325.
%H Hacène Belbachir, Amine Belkhir, and Ihab-Eddin Djellas, <a href="https://digitalcommons.pvamu.edu/aam/vol17/iss2/15/">Permanent of Toeplitz-Hessenberg Matrices with Generalized Fibonacci and Lucas entries</a>, Applications and Applied Mathematics: An International Journal (AAM 2022), Vol. 17, Iss. 2, Art. 15, 558-570.
%H P. Bhadouria, D. Jhala, and B. Singh, <a href="http://dx.doi.org/10.22436/jmcs.08.01.07">Binomial Transforms of the k-Lucas Sequences and its Properties</a>, The Journal of Mathematics and Computer Science (JMCS), Volume 8, Issue 1, Pages 81-92; sequence L_{2,n}.
%H Abdullah Çağman, <a href="https://doi.org/10.18514/MMN.2023.4143">Repdigits as sums of three Half-companion Pell numbers</a>, Miskolc Mathematical Notes (Hungary, 2023) Vol. 24, No. 2, 687-697, MMN-4143.
%H Kwang-Wu Chen and Yu-Ren Pan, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Pan/pan32.html">Greatest Common Divisors of Shifted Horadam Sequences</a>, J. Int. Seq., Vol. 23 (2020), Article 20.5.8.
%H Robert Dougherty-Bliss, <a href="https://sites.math.rutgers.edu/~zeilberg/Theses/RobertDoughertyBlissThesis.pdf">Experimental Methods in Number Theory and Combinatorics</a>, Ph. D. Dissertation, Rutgers Univ. (2024). See p. 38.
%H Sergio Falcon, <a href="https://web.archive.org/web/20220701035124/http://saspublisher.com/wp-content/uploads/2014/06/SJET24C669-675.pdf">On The Generating Functions of the Powers of the K-Fibonacci Numbers</a>, Scholars Journal of Engineering and Technology (SJET), 2014; 2 (4C):669-675. [Wayback Machine link]
%H Bakir Farhi, <a href="https://www.emis.de/journals/JIS/VOL22/Farhi/farhi19.html">Summation of Certain Infinite Lucas-Related Series</a>, J. Int. Seq., Vol. 22 (2019), Article 19.1.6.
%H Bernadette Faye, and Florian Luca, <a href="http://arxiv.org/abs/1508.05714">Pell Numbers whose Euler Function is a Pell Number</a>, arXiv:1508.05714 [math.NT], 2015.
%H M. Cetin Firengiz and A. Dil, <a href="http://www.nntdm.net/papers/nntdm-20/NNTDM-20-4-21-32.pdf">Generalized Euler-Seidel method for second order recurrence relations</a>, Notes on Number Theory and Discrete Mathematics, Vol. 20, 2014, No. 4, 21-32.
%H Taras Goy and Mark Shattuck, <a href="https://doi.org/10.2478/amsil-2023-0027">Determinants of Toeplitz-Hessenberg Matrices with Generalized Leonardo Number Entries</a>, Ann. Math. Silesianae (2023). See p. 3.
%H V. E. Hoggatt, Jr., and G. L. Alexanderson, <a href="https://www.fq.math.ca/Scanned/14-2/hoggatt1.pdf">Sums of Partition Sets in Generalized Pascal Triangles I</a>, The Fibonacci Quarterly, Vol. 14, No. 2 (1976), pp. 117-125.
%H Refik Keskin and Olcay Karaatli, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Karaatli/karaatli5.html">Some New Properties of Balancing Numbers and Square Triangular Numbers</a>, Journal of Integer Sequences, Vol. 15 (2012), Article 12.1.4.
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>.
%H Edouard Lucas, <a href="https://doi.org/10.2307/2369308">Théorie des Fonctions Numériques Simplement Périodiques</a>, Amer. J. Math., 1 (1878), 184-240.
%H Edouard Lucas, <a href="http://edouardlucas.free.fr/oeuvres/Theorie_des_fonctions_simplement_periodiques.pdf">Théorie des Fonctions Numériques Simplement Périodiques</a>, I", Amer. J. Math., 1 (1878), 184-240 and 289-321.
%H Edouard Lucas, <a href="http://www.mathstat.dal.ca/FQ/Books/Complete/simply-periodic.pdf">The Theory of Simply Periodic Numerical Functions</a>, Fibonacci Association, 1969. English translation of article "Théorie des Fonctions Numériques Simplement Périodiques, I", Amer. J. Math., 1 (1878), 184-240.
%H aBa Mbirika, Janeè Schrader, and Jürgen Spilker, <a href="https://arxiv.org/abs/2301.05758">Pell and associated Pell braid sequences as GCDs of sums of k consecutive Pell, balancing, and related numbers</a>, arXiv:2301.05758 [math.NT], 2023. See also <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Mbirika/mbir5.html">J. Int. Seq.</a> (2023) Vol. 26, Art. 23.6.4.
%H Ezgi Kantarcı Oguz, Cem Yalım Özel, and Mohan Ravichandran, <a href="https://www.mat.univie.ac.at/~slc/wpapers/FPSAC2023/67.pdf">Chainlink Polytopes</a>, Séminaire Lotharingien de Combinatoire, Proc. 35th Conf. Formal Power Series and Alg. Comb. (Davis, 2023) Vol. 89B, Art. #67.
%H Hideyuki Ohtsuka, <a href="https://www.jstor.org/stable/48662478">Problem 12090</a>, The American Mathematical Monthly, Vol. 126, No. 2 (2019), p. 180; <a href="https://www.jstor.org/stable/45401321">A Pell-Lucas Computation of Pi</a>, Solution to Problem 12090 by M. Vowe, ibid., Vol. 127, No. 7 (2020), pp. 666-667.
%H Neşe Ömür, Gökhan Soydan, Yücel Türker Ulutaş, and Yusuf Doğru, <a href="https://doi.org/10.21857/ydkx2cwq49">On triangles with coordinates of vertices from the terms of the sequences {U_kn} and {V_kn}</a>, Matematičke Znanosti, Vol. 24 = 542(2020), 15-27.
%H Arzu Özkoç, <a href="http://link.springer.com/article/10.1186/s13662-015-0486-7/fulltext.html">Some algebraic identities on quadra Fibona-Pell integer sequence</a>, Advances in Difference Equations, 2015, 2015:148.
%H Serge Perrine, <a href="http://article.scirea.org/pdf/11150.pdf">About the diophantine equation z^2 = 32y^2 - 16</a>, SCIREA Journal of Mathematics (2019) Vol. 4, Issue 5, 126-139.
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.
%H Salah Eddine Rihane and Alain Togbé, <a href="https://doi.org/10.33039/ami.2021.03.014">On the intersection of Padovan, Perrin sequences and Pell, Pell-Lucas sequences</a>, Annales Mathematicae et Informaticae (2021).
%H Yüksel Soykan, <a href="https://doi.org/10.9734/AIR/2019/v20i230154">On Summing Formulas For Generalized Fibonacci and Gaussian Generalized Fibonacci Numbers</a>, Advances in Research (2019) Vol. 20, No. 2, 1-15, Article AIR.51824.
%H Yüksal Soykan, <a href="https://doi.org/10.9734/AJARR/2020/v8i130192">On Summing Formulas for Horadam Numbers</a>, Asian Journal of Advanced Research and Reports (2020) Vol. 8, Issue 1, 45-61.
%H Yüksel Soykan, <a href="https://doi.org/10.9734/jamcs/2020/v35i130241">Generalized Fibonacci Numbers: Sum Formulas</a>, Journal of Advances in Mathematics and Computer Science (2020) Vol. 35, No. 1, 89-104.
%H Yüksel Soykan, <a href="https://doi.org/10.9734/AJARR/2020/v9i130212">Closed Formulas for the Sums of Squares of Generalized Fibonacci Numbers</a>, Asian Journal of Advanced Research and Reports (2020) Vol. 9, No. 1, 23-39, Article no. AJARR.55441.
%H Yüksel Soykan, <a href="https://doi.org/10.34198/ejms.4220.297331">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</a>, Earthline Journal of Mathematical Sciences (2020) Vol. 4, No. 2, 297-331.
%H Yüksel Soykan, <a href="http://www.ijaamm.com/uploads/2/1/4/8/21481830/v8n1p1_1-14.pdf">On Generalized (r, s)-numbers</a>, Int. J. Adv. Appl. Math. and Mech. (2020) Vol. 8, No. 1, 1-14.
%H Yüksel Soykan, Mehmet Gümüş, and Melih Göcen, <a href="https://doi.org/10.13140/RG.2.2.21008.97289">A Study On Dual Hyperbolic Generalized Pell Numbers</a>, Zonguldak Bülent Ecevit University (Zonguldak, Turkey, 2019).
%H Robin James Spivey, <a href="https://doi.org/10.7546/nntdm.2019.25.3.170-184">Close encounters of the golden and silver ratios</a>, Notes on Number Theory and Discrete Mathematics (2019) Vol. 25, No. 3, 170-184.
%H Anetta Szynal-Liana, Iwona Włoch, and Mirosław Liana, <a href="https://doi.org/10.17951/a.2022.76.2.33-44">Generalized commutative quaternion polynomials of the Fibonacci type</a>, Annales Math. Sect. A, Univ. Mariae Curie-Skłodowska (Poland 2022) Vol. 76, No. 2, 33-44.
%H A. Tekcan, M. Tayat, and M. E. Ozbek, <a href="http://dx.doi.org/10.1155/2014/897834">The diophantine equation 8x^2-y^2+8x(1+t)+(2t+1)^2=0 and t-balancing numbers</a>, ISRN Combinatorics, Volume 2014, Article ID 897834, 5 pages.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/IndependentEdgeSet.html">Independent Edge Set</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Matching.html">Matching</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PellNumber.html">Pell Number</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SunletGraph.html">Sunlet Graph</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Lucas_sequence#Specific_names">Lucas sequence</a>.
%H Zongzhen Xie, Hanpeng Gao, and Zhaoyong Huang, <a href="http://maths.nju.edu.cn/~huangzy/tiltausl.pdf">Tilting modules over Auslander algebras of Nakayama algebras with radical cube zero</a>, Nanjing University (China, 2020).
%H Fatih Yılmaz and Mustafa Özkan, <a href="https://doi.org/10.3390/axioms11060255">On the Generalized Gaussian Fibonacci Numbers and Horadam Hybrid Numbers: A Unified Approach</a>, Axioms (2022) Vol. 11, No. 6, 255.
%H Abdelmoumène Zekiri, Farid Bencherif, and Rachid Boumahdi, <a href="https://www.emis.de/journals/JIS/VOL21/Zekiri/zekiri4.html">Generalization of an Identity of Apostol</a>, J. Int. Seq., Vol. 21 (2018), Article 18.5.1.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,1).
%H <a href="/index/Lu#Lucas">Index entries for Lucas sequences</a>.
%H <a href="/index/Rea#recur1">Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)</a>.
%F a(n) = 2 * A001333(n).
%F a(n) = A100227(n) + 1.
%F O.g.f.: (2 - 2*x)/(1 - 2*x - x^2). - _Simon Plouffe_ in his 1992 dissertation
%F a(n) = (1 + sqrt(2))^n + (1 - sqrt(2))^n. - Mario Catalani (mario.catalani(AT)unito.it), Mar 17 2003
%F a(n) = A000129(2*n)/A000129(n), n > 0. - _Paul Barry_, Feb 06 2004
%F From _Miklos Kristof_, Mar 19 2007: (Start)
%F Given F(n) = A000129(n), the Pell numbers, and L(n) = a(n), then:
%F L(n+m) + (-1)^m*L(n-m) = L(n)*L(m).
%F L(n+m) - (-1)^m*L(n-m) = 8*F(n)*F(m).
%F 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).
%F 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).
%F 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).
%F 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).
%F (End)
%F a(n) = 2*(A000129(n+1) - A000129(n)). - _R. J. Mathar_, Nov 16 2007
%F 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
%F a(n) = [x^n] ( (1 + 2*x + sqrt(1 + 4*x + 8*x^2))/2 )^n for n >= 1. - _Peter Bala_, Jun 23 2015
%F From _Kai Wang_, Jan 14 2020: (Start)
%F A000129(m - n) = ((-1)^n * (A000129(m) * a((n) - a((m) * A000129(n))/2.
%F A000129(m + n) = (A000129(m) * a((n) + a((m)*A000129(n))/2.
%F a(n)^2 - a(n + 1) * a(n - 1) = (-1)^(n) * 8.
%F a(n)^2 - a(n + r) * a(n - r) = (-1)^(n - r - 1) * 8 * A000129(r)^2.
%F a(m) * a(n + 1) - a(m + 1) * a(n) = (-1)^(n - 1) * 8 * A000129(m - n).
%F a(m - n) = (-1)^(n) * (a(m) * a(n) - 8 * A000129(m) * A000129(n))/2.
%F a(m + n) = (a(m) * a(n) + 8 * A000129(m) * A000129(n))/2.
%F (End)
%F E.g.f.: 2*exp(x)*cosh(sqrt(2)*x). - _Stefano Spezia_, Jan 15 2020
%F a(n) = A000129(n+1) + A000129(n-1) for n>0 with a(0)=2. - _Rigoberto Florez_, Jul 12 2020
%F a(n) = (-1)^n * (a(n)^3 - a(3*n))/3. - _Greg Dresden_, Jun 16 2021
%F a(n) = (a(n+2) + a(n-2))/6 for n >= 2. - _Greg Dresden_, Jun 23 2021
%F From _Greg Dresden_ and _Tongjia Rao_, Sep 09 2021: (Start)
%F a(3n+2)/a(3n-1) = [14, ..., 14, -3] with (n+1) 14's.
%F a(3n+3)/a( 3n ) = [14, ..., 14, 7] with n 14's.
%F a(3n+4)/a(3n+1) = [14, ..., 14, 17] with n 14's. (End)
%F From _Peter Bala_, Nov 16 2022: (Start)
%F a(n) = trace([2, 1; 1, 0]^n) for n >= 1.
%F 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.
%F a(3^n) == A271222(n) (mod 3^n). (End)
%F Sum_{n>=1} arctan(2/a(n))*arctan(2/a(n+1)) = Pi^2/32 (A244854) (Ohtsuka, 2019). - _Amiram Eldar_, Feb 11 2024
%p A002203 := proc(n)
%p option remember;
%p if n <= 1 then
%p 2;
%p else
%p 2*procname(n-1)+procname(n-2) ;
%p end if;
%p end proc: # _R. J. Mathar_, May 11 2013
%p # second Maple program:
%p a:= n-> (<<0|1>, <1|2>>^n. <<2, 2>>)[1, 1]:
%p seq(a(n), n=0..30); # _Alois P. Heinz_, Jan 26 2018
%p a := n -> 2*I^n*ChebyshevT(n, -I):
%p seq(simplify(a(n)), n = 0..30); # _Peter Luschny_, Dec 03 2023
%t Table[LucasL[n, 2], {n, 0, 30}] (* _Zerinvary Lajos_, Jul 09 2009 *)
%t LinearRecurrence[{2, 1}, {2, 2}, 50] (* _Vincenzo Librandi_, Aug 15 2015 *)
%t Table[(1 - Sqrt[2])^n + (1 + Sqrt[2])^n, {n, 0, 20}] // Expand (* _Eric W. Weisstein_, Oct 03 2017 *)
%t LucasL[Range[0, 20], 2] (* _Eric W. Weisstein_, Oct 03 2017 *)
%t CoefficientList[Series[(2 (1 - x))/(1 - 2 x - x^2), {x, 0, 20}], x] (* _Eric W. Weisstein_, Oct 03 2017 *)
%o (Sage) [lucas_number2(n,2,-1) for n in range(0, 29)] # _Zerinvary Lajos_, Apr 30 2009
%o (Haskell)
%o a002203 n = a002203_list !! n
%o a002203_list =
%o 2 : 2 : zipWith (+) (map (* 2) $ tail a002203_list) a002203_list
%o -- _Reinhard Zumkeller_, Oct 03 2011
%o (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
%o (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
%o (PARI) a(n) = my(w=quadgen(8)); (1+w)^n + (1-w)^n; \\ _Michel Marcus_, Jun 17 2021
%Y Equals twice A001333.
%Y Cf. A000129, A100227, A244854.
%Y Bisections are A003499 and A077444.
%K nonn,easy
%O 0,1
%A _N. J. A. Sloane_
%E More terms from Larry Reeves (larryr(AT)acm.org), Dec 03 2001
| 