%I M3420 N1384 #379 Aug 22 2024 17:37:54

%S 1,4,11,29,76,199,521,1364,3571,9349,24476,64079,167761,439204,

%T 1149851,3010349,7881196,20633239,54018521,141422324,370248451,

%U 969323029,2537720636,6643838879,17393796001,45537549124,119218851371,312119004989,817138163596,2139295485799

%N Bisection of Lucas sequence: a(n) = L(2*n+1).

%C In any generalized Fibonacci sequence {f(i)}, Sum_{i=0..4n+1} f(i) = a(n)*f(2n+2). - _Lekraj Beedassy_, Dec 31 2002

%C The continued fraction expansion for F((2n+1)*(k+1))/F((2n+1)*k), k>=1 is [a(n),a(n),...,a(n)] where there are exactly k elements (F(n) denotes the n-th Fibonacci number). E.g., continued fraction for F(12)/F(9) is [4, 4,4]. - _Benoit Cloitre_, Apr 10 2003

%C See A135064 for a possible connection with Galois groups of quintics.

%C Sequence of all positive integers k such that continued fraction [k,k,k,k,k,k,...] belongs to Q(sqrt(5)). - _Thomas Baruchel_, Sep 15 2003

%C All positive integer solutions of Pell equation a(n)^2 - 5*b(n)^2 = -4 together with b(n)=A001519(n), n>=0.

%C a(n) = L(n,-3)*(-1)^n, where L is defined as in A108299; see also A001519 for L(n,+3).

%C Inverse binomial transform of A030191. - _Philippe Deléham_, Oct 04 2005

%C General recurrence is a(n) = (a(1)-1)*a(n-1) - a(n-2), a(1) >= 4, lim_{n->infinity} a(n) = x*(k*x+1)^n, k =(a(1)-3), x=(1+sqrt((a(1)+1)/(a(1)-3)))/2. Examples in OEIS: a(1)=4 gives A002878. a(1)=5 gives A001834. a(1)=6 gives A030221. a(1)=7 gives A002315. a(1)=8 gives A033890. a(1)=9 gives A057080. a(1)=10 gives A057081. - _Ctibor O. Zizka_, Sep 02 2008

%C Let r = (2n+1), then a(n), n>0 = Product_{k=1..floor((r-1)/2)} (1 + sin^2 k*Pi/r); e.g., a(3) = 29 = (3.4450418679...)*(4.801937735...)*(1.753020396...). - _Gary W. Adamson_, Nov 26 2008

%C a(n+1) is the Hankel transform of A001700(n)+A001700(n+1). - _Paul Barry_, Apr 21 2009

%C a(n) is equal to the permanent of the (2n) X (2n) tridiagonal matrix with sqrt(5)'s along the main diagonal, i's along the superdiagonal and the subdiagonal (i is the imaginary unit), and 0's everywhere else. - _John M. Campbell_, Jun 09 2011

%C Conjecture: for n > 0, a(n) = sqrt(Fibonacci(4*n+3) + Sum_{k=2..2*n} Fibonacci(2*k)). - _Alex Ratushnyak_, May 06 2012

%C Pisano period lengths: 1, 3, 4, 3, 2, 12, 8, 6, 12, 6, 5, 12, 14, 24, 4, 12, 18, 12, 9, 6, ... . - _R. J. Mathar_, Aug 10 2012

%C The continued fraction [a(n); a(n), a(n), ...] = phi^(2n+1), where phi is the golden ratio, A001622. - _Thomas Ordowski_, Jun 05 2013

%C Solutions (x, y) = (a(n), a(n+1)) satisfying x^2 + y^2 = 3xy + 5. - _Michel Lagneau_, Feb 01 2014

%C Conjecture: except for the number 3, a(n) are the numbers such that a(n)^2+2 are Lucas numbers. - _Michel Lagneau_, Jul 22 2014

%C Comment on the preceding conjecture: It is clear that all a(n) satisfy a(n)^2 + 2 = L(2*(2*n+1)) due to the identity (17 c) of Vajda, p. 177: L(2*n) + 2*(-1)^n = L(n)^2 (take n -> 2*n+1). - _Wolfdieter Lang_, Oct 10 2014

%C Limit_{n->oo} a(n+1)/a(n) = phi^2 = phi + 1 = (3+sqrt(5))/2. - _Derek Orr_, Jun 18 2015

%C If d[k] denotes the sequence of k-th differences of this sequence, then d[0](0), d[1](1), d[2](2), d[3](3), ... = A048876, cf. message to SeqFan list by P. Curtz on March 2, 2016. - _M. F. Hasler_, Mar 03 2016

%C a(n-1) and a(n) are the least phi-antipalindromic numbers (A178482) with 2*n and 2*n+1 digits in base phi, respectively. - _Amiram Eldar_, Jul 07 2021

%C Triangulate (hyperbolic) 2-space such that around every vertex exactly 7 triangles touch. Call any 7 triangles having a common vertex the first layer and let the (n+1)-st layer be all triangles that do not appear in any of the first n layers and have a common vertex with the n-th layer. Then the n-th layer contains 7*a(n-1) triangles. E.g., the first layer (by definition) contains 7 triangles, the second layer (the "ring" of triangles around the first layer) consists of 28 triangles, the third layer (the next "ring") consists of 77 triangles, and so on. - _Nicolas Nagel_, Aug 13 2022

%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).

%D Steven Vajda, Fibonacci and Lucas numbers and the Golden Section, Ellis Horwood Ltd., Chichester, 1989.

%H Vincenzo Librandi, <a href="/A002878/b002878.txt">Table of n, a(n) for n = 0..200</a>

%H Marco Abrate, Stefano Barbero, Umberto Cerruti and Nadir Murru, <a href="https://www.emis.de/journals/INTEGERS/papers/p38/p38.Abstract.html">Polynomial sequences on quadratic curves</a>, Integers, Vol. 15, 2015, #A38.

%H Kasper K. S. Andersen, Lisa Carbone and Diego Penta, <a href="https://pdfs.semanticscholar.org/8f0c/c3e68d388185129a56ed73b5d21224659300.pdf">Kac-Moody Fibonacci sequences, hyperbolic golden ratios, and real quadratic fields</a>, Journal of Number Theory and Combinatorics, Vol 2, No. 3 pp 245-278, 2011. See Section 9.

%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Barry3/barry132.html">On the Central Coefficients of Bell Matrices</a>, J. Int. Seq., Vol. 14 (2011), Article 11.4.3, page 9.

%H Hacène Belbachir, Soumeya Merwa Tebtoub and László Németh, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Nemeth/nemeth7.html">Ellipse Chains and Associated Sequences</a>, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.

%H Joshua P. Bowman, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL27/Bowman/bowman4.html">Compositions with an Odd Number of Parts, and Other Congruences</a>, J. Int. Seq (2024) Vol. 27, Art. 24.3.6. See p. 25.

%H Nathan D. Cahill, John R. D'Errico and John P. Spence, <a href="http://www.fq.math.ca/Scanned/41-1/cahill.pdf">Complex Factorizations of the Fibonacci and Lucas Numbers</a>, Fibonacci Quarterly, 1(41):13-19, 2003.

%H L. Carlitz, <a href="https://fq.math.ca/Scanned/5-1/elementary5-1.pdf">Problem B-110</a>, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 5, No. 1 (1967), p. 108; <a href="https://fq.math.ca/Scanned/5-5/elementary5-5.pdf">An Infinite Series Equality</a>, Solution to Problem B-110 by the proposer, ibid., Vol. 5, No. 5 (1967), pp. 469-470.

%H Paul Curtz, <a href="http://list.seqfan.eu/oldermail/seqfan/2016-March/016180.html">A269500</a>, SeqFan list, March 2, 2016.

%H Murray Elder and Arkadius Kalka, <a href="https://arxiv.org/abs/1310.0933">Logspace computations for rigid Garside groups</a>, arXiv preprint arXiv:1310.0933 [math.GR], 2013.

%H Sergio Falcon, <a href="https://doi.org/10.4236/am.2014.515216">Relationships between Some k-Fibonacci Sequences</a>, Applied Mathematics, Vol. 5 (2014), pp. 2226-2234.

%H Alex Fink, Richard K. Guy and Mark Krusemeyer, <a href="https://cdm.ucalgary.ca/article/view/61940">Partitions with parts occurring at most thrice</a>, Contributions to Discrete Mathematics, Vol. 3, No. 2 (2008), pp. 76-114. See Section 13.

%H Dale Gerdemann, <a href="https://www.youtube.com/watch?vEQYQ4bEUX34">Collision of Digits</a> "Also interesting are the two bisections of the Lucas numbers A005248 (digit minimizer) and A002878 (digit maximizer). I particularly like the multiples of A005248 because I have this image of the two digits piling up on top of each other and then spreading out like waves".

%H André Gougenheim, About the linear sequence of integers such that each term is the sum of the two preceding <a href="http://www.fq.math.ca/Scanned/9-3/gougenheim-a.pdf">Part 1</a> <a href="http://www.fq.math.ca/Scanned/9-3/gougenheim-b.pdf">Part 2</a>, Fib. Quart., Vol. 9, No. 3 (1971), pp. 277-295, 298.

%H Tian-Xiao He and Louis W. Shapiro, <a href="https://doi.org/10.1016/j.laa.2017.06.025">Fuss-Catalan matrices, their weighted sums, and stabilizer subgroups of the Riordan group</a>, Lin. Alg. Applic., Vol. 532 (2017) pp. 25-41, example p 34.

%H Seong Ju Kim, Ryan Stees and Laura Taalman, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Stees/stees4.html">Sequences of Spiral Knot Determinants</a>, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.4.

%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>.

%H Ioana-Claudia Lazăr, <a href="https://arxiv.org/abs/1904.06555">Lucas sequences in t-uniform simplicial complexes</a>, arXiv:1904.06555 [math.GR], 2019.

%H D. H. Lehmer, <a href="https://doi.org/10.1090/S0002-9904-1943-07880-9">Recurrence formulas for certain divisor functions</a>, Bull. Amer. Math. Soc., Vol. 49, No. 2 (1943), pp. 150-156.

%H Giovanni Lucca, <a href="https://forumgeom.fau.edu/FG2019volume19/FG201902index.html">Integer Sequences and Circle Chains Inside a Hyperbola</a>, Forum Geometricorum, Vol. 19 (2019), pp. 11-16.

%H Donatella Merlini and Renzo Sprugnoli, <a href="https://doi.org/10.1016/j.disc.2016.08.017">Arithmetic into geometric progressions through Riordan arrays</a>, Discrete Mathematics, Vol. 340, No. 2 (2017), pp. 160-174.

%H Prabha Sivaraman Nair and Rejikumar Karunakaran, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL27/Nair/nair11.html">On k-Fibonacci Brousseau Sums</a>, J. Int. Seq. (2024) Art. No. 24.6.4. See p. 3.

%H Serge Perrine, <a href="https://www.fq.math.ca/Papers1/54-2/Perrine02242016.pdf">Some properties of the equation x^2=5y^2-4</a>, The Fibonacci Quarterly, Vol. 54, No. 2 (2016) pp. 172-177.

%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 Ryan Stees, <a href="https://commons.lib.jmu.edu/honors201019/84">Sequences of Spiral Knot Determinants</a>, Senior Honors Projects, Paper 84, James Madison Univ., May 2016.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FibonacciPolynomial.html">Fibonacci Polynomial</a>.

%H H. C. Williams and R. K. Guy, <a href="https://doi.org/10.1142/S1793042111004587">Some fourth-order linear divisibility sequences</a>, Intl. J. Number Theory, Vol. 7, No. 5 (2011), pp. 1255-1277.

%H H. C. Williams and R. K. Guy, <a href="https://www.emis.de/journals/INTEGERS/papers/a17self/a17self.Abstract.html">Some Monoapparitic Fourth Order Linear Divisibility Sequences</a>, Integers, Volume 12A (2012), The John Selfridge Memorial Volume.

%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials</a>.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1).

%F a(n+1) = 3*a(n) - a(n-1).

%F G.f.: (1+x)/(1-3*x+x^2). - _Simon Plouffe_ in his 1992 dissertation

%F a(n) = S(2*n, sqrt(5)) = S(n, 3) + S(n-1, 3); S(n, x) := U(n, x/2), Chebyshev polynomials of 2nd kind, A049310. S(n, 3) = A001906(n+1) (even-indexed Fibonacci numbers).

%F a(n) ~ phi^(2*n+1). - Joe Keane (jgk(AT)jgk.org), May 15 2002

%F Let q(n, x) = Sum_{i=0..n} x^(n-i)*binomial(2*n-i, i); then (-1)^n*q(n, -1) = a(n). - _Benoit Cloitre_, Nov 10 2002

%F a(n) = A005248(n+1) - A005248(n) = -1 + Sum_{k=0..n} A005248(k). - _Lekraj Beedassy_, Dec 31 2002

%F a(n) = 2^(-n)*A082762(n) = 4^(-n)*Sum_{k>=0} binomial(2*n+1, 2*k)*5^k; see A091042. - _Philippe Deléham_, Mar 01 2004

%F a(n) = (-1)^n*Sum_{k=0..n} (-5)^k*binomial(n+k, n-k). - _Benoit Cloitre_, May 09 2004

%F From _Paul Barry_, May 27 2004: (Start)

%F Both bisection and binomial transform of A000204.

%F a(n) = Fibonacci(2n) + Fibonacci(2n+2). (End)

%F Sequence lists the numerators of sinh((2*n-1)*psi) where the denominators are 2; psi=log((1+sqrt(5))/2). Offset 1. a(3)=11. - Al Hakanson (hawkuu(AT)gmail.com), Mar 25 2009

%F a(n) = A001906(n) + A001906(n+1). - _Reinhard Zumkeller_, Jan 11 2012

%F a(n) = floor(phi^(2n+1)), where phi is the golden ratio, A001622. - _Thomas Ordowski_, Jun 10 2012

%F a(n) = A014217(2*n+1) = A014217(2*n+2) - A014217(2*n). - _Paul Curtz_, Jun 11 2013

%F Sum_{n >= 0} 1/(a(n) + 5/a(n)) = 1/2. Compare with A005248, A001906, A075796. - _Peter Bala_, Nov 29 2013

%F a(n) = lim_{m->infinity} Fibonacci(m)^(4n+1)*Fibonacci(m+2*n+1)/ Sum_{k=0..m} Fibonacci(k)^(4n+2). - _Yalcin Aktar_, Sep 02 2014

%F From _Peter Bala_, Mar 22 2015: (Start)

%F The aerated sequence (b(n))n>=1 = [1, 0, 4, 0, 11, 0, 29, 0, ...] is a fourth-order linear divisibility sequence; that is, if n | m then b(n) | b(m). It is the case P1 = 0, P2 = -1, Q = -1 of the 3-parameter family of divisibility sequences found by Williams and Guy.

%F b(n) = (1/2)*((-1)^n - 1)*F(n) + (1 + (-1)^(n-1))*F(n+1), where F(n) is a Fibonacci number. The o.g.f. is x*(1 + x^2)/(1 - 3*x^2 + x^4).

%F Exp( Sum_{n >= 1} 2*b(n)*x^n/n ) = 1 + Sum_{n >= 1} 2*F(n)*x^n.

%F Exp( Sum_{n >= 1} (-2)*b(n)*x^n/n ) = 1 + Sum_{n >= 1} 2*F(n)*(-x)^n.

%F Exp( Sum_{n >= 1} 4*b(n)*x^n/n ) = 1 + Sum_{n >= 1} 4*A029907(n)*x^n.

%F Exp( Sum_{n >= 1} (-4)*b(n)*x^n/n ) = 1 + Sum_{n >= 1} 4*A029907(n)*(-x)^n. Cf. A002315, A004146, A113224 and A192425. (End)

%F a(n) = sqrt(5*F(2*n+1)^2-4), where F(n) = A000045(n). - _Derek Orr_, Jun 18 2015

%F For n > 1, a(n) = 5*F(2*n-1) + L(2*n-3) with F(n) = A000045(n). - _J. M. Bergot_, Oct 25 2015

%F For n > 0, a(n) = L(n-1)*L(n+2) + 4*(-1)^n. - _J. M. Bergot_, Oct 25 2015

%F For n > 2, a(n) = a(n-2) + F(n+2)^2 + F(n-3)^2 = L(2*n-3) + F(n+2)^2 + F(n-3)^2. - _J. M. Bergot_, Feb 05 2016 and Feb 07 2016

%F E.g.f.: ((sqrt(5) - 5)*exp((3-sqrt(5))*x/2) + (5 + sqrt(5))*exp((3+sqrt(5))*x/2))/(2*sqrt(5)). - _Ilya Gutkovskiy_, Apr 24 2016

%F a(n) = Sum_{k=0..n} (-1)^floor(k/2)*binomial(n-floor((k+1)/2), floor(k/2))*3^(n-k). - _L. Edson Jeffery_, Feb 26 2018

%F a(n)*F(m+2n-1) = F(m+4n-2)-F(m), with Fibonacci number F(m), empirical observation. - _Dan Weisz_, Jul 30 2018

%F a(n) = -a(-1-n) for all n in Z. - _Michael Somos_, Jul 31 2018

%F Sum_{n>=0} 1/a(n) = A153416. - _Amiram Eldar_, Nov 11 2020

%F a(n) = Product_{k=1..n} (1 + 4*sin(2*k*Pi/(2*n+1))^2). - _Seiichi Manyama_, Apr 30 2021

%F Sum_{n>=0} (-1)^n/a(n) = (1/sqrt(5)) * A153387 (Carlitz, 1967). - _Amiram Eldar_, Feb 05 2022

%F The continued fraction [a(n);a(n),a(n),...] = phi^(2*n+1), with phi = A001622. - _A.H.M. Smeets_, Feb 25 2022

%F a(n) = 2*sinh((2*n + 1)*arccsch(2)). - _Peter Luschny_, May 25 2022

%F This gives the sequence with 2 1's prepended: b(1)=b(2)=1 and, for k >= 3, b(k) = Sum_{j=1..k-2} (2^(k-j-1) - 1)*b(j). - _Neal Gersh Tolunsky_, Oct 28 2022 (formula due to Jon E. Schoenfield)

%F For n > 0, a(n) = 1 + 1/(Sum_{k>=1} F(k)/phi^(2*n*k + k)). - _Diego Rattaggi_, Nov 08 2023

%e G.f. = 1 + 4*x + 11*x^2 + 29*x^3 + 76*x^4 + 199*x^5 + 521*x^6 + ... - _Michael Somos_, Jan 13 2019

%p A002878 := proc(n)

%p option remember;

%p if n <= 1 then

%p op(n+1,[1,4]);

%p else

%p 3*procname(n-1)-procname(n-2) ;

%p end if;

%p end proc: # _R. J. Mathar_, Apr 30 2017

%t a[n_]:= FullSimplify[GoldenRatio^n - GoldenRatio^-n]; Table[a[n], {n, 1, 40, 2}]

%t a[1]=1; a[2]=4; a[n_]:=a[n]= 3a[n-1] -a[n-2]; Array[a, 40]

%t LinearRecurrence[{3, -1}, {1, 4}, 41] (* _Jean-François Alcover_, Sep 23 2017 *)

%t Table[Sum[(-1)^Floor[k/2] Binomial[n -Floor[(k+1)/2], Floor[k/2]] 3^(n - k), {k, 0, n}], {n, 0, 40}] (* _L. Edson Jeffery_, Feb 26 2018 *)

%t a[ n_] := Fibonacci[2n] + Fibonacci[2n+2]; (* _Michael Somos_, Jul 31 2018 *)

%t a[ n_]:= LucasL[2n+1]; (* _Michael Somos_, Jan 13 2019 *)

%o (Magma) [Lucas(2*n+1): n in [0..40]]; // _Vincenzo Librandi_, Apr 16 2011

%o (Haskell)

%o a002878 n = a002878_list !! n

%o a002878_list = zipWith (+) (tail a001906_list) a001906_list

%o -- _Reinhard Zumkeller_, Jan 11 2012

%o (PARI) a(n)=fibonacci(2*n)+fibonacci(2*n+2) \\ _Charles R Greathouse IV_, Jun 16 2011

%o (PARI) for(n=1,40,q=((1+sqrt(5))/2)^(2*n-1);print1(contfrac(q)[1],", ")) \\ _Derek Orr_, Jun 18 2015

%o (PARI) Vec((1+x)/(1-3*x+x^2) + O(x^40)) \\ _Altug Alkan_, Oct 26 2015

%o (Sage) [lucas_number2(2*n+1,1,-1) for n in (0..40)] # _G. C. Greubel_, Jul 15 2019

%o (GAP) List([0..40], n-> Lucas(1,-1,2*n+1)[2] ); # _G. C. Greubel_, Jul 15 2019

%o (Python)

%o a002878 = [1, 4]

%o for n in range(30): a002878.append(3*a002878[-1] - a002878[-2])

%o print(a002878) # _Gennady Eremin_, Feb 05 2022

%Y Cf. A000204. a(n) = A060923(n, 0), a(n)^2 = A081071(n).

%Y Cf. A005248 [L(2n) = bisection (even n) of Lucas sequence].

%Y Cf. A001906 [F(2n) = bisection (even n) of Fibonacci sequence], A000045, A002315, A004146, A029907, A113224, A153387, A153416, A178482, A192425, A285992 (prime subsequence).

%Y Cf. similar sequences of the type k*F(n)*F(n+1)+(-1)^n listed in A264080.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_

%E Chebyshev and Pell comments from _Wolfdieter Lang_, Aug 31 2004