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Alternate Lucas numbers - 2.
(Formerly M3867)
37

%I M3867 #241 Mar 28 2024 21:57:08

%S 0,1,5,16,45,121,320,841,2205,5776,15125,39601,103680,271441,710645,

%T 1860496,4870845,12752041,33385280,87403801,228826125,599074576,

%U 1568397605,4106118241,10749957120,28143753121,73681302245

%N Alternate Lucas numbers - 2.

%C This is the r=5 member in the r-family of sequences S_r(n) defined in A092184 where more information can be found.

%C Number of spanning trees of the wheel W_n on n+1 vertices. - _Emeric Deutsch_, Mar 27 2005

%C Also number of spanning trees of the n-helm graph. - _Eric W. Weisstein_, Jul 16 2011

%C a(n) is the smallest number requiring n terms when expressed as a sum of Lucas numbers (A000204). - _David W. Wilson_, Jan 10 2006

%C This sequence has a primitive prime divisor for all terms beyond the twelfth. - Anthony Flatters (Anthony.Flatters(AT)uea.ac.uk), Aug 17 2007

%C From _Giorgio Balzarotti_, Mar 11 2009: (Start)

%C Determinant of power series of gamma matrix with determinant 1:

%C a(n) = Determinant(A + A^2 + A^3 + A^4 + A^5 + ... + A^n)

%C where A is the submatrix A(1..2,1..2) of the matrix with factorial determinant

%C A = [[1,1,1,1,1,1,...],[1,2,1,2,1,2,...],[1,2,3,1,2,3,...],[1,2,3,4,1,2,...],

%C [1,2,3,4,5,1,...],[1,2,3,4,5,6,...],...]. Note: Determinant A(1..n,1..n)= (n-1)!.

%C See A158039, A158040, A158041, A158042, A158043, A158044, for sequences of matrix 2!,3!,... (End)

%C The previous comment could be rephrased as: a(n) = -det(A^n - I) where I is the 2 X 2 identity matrix and A = [1, 1; 1, 2]. - _Peter Bala_, Mar 20 2015

%C a(n) is also the number of points of Arnold's "cat map" that are on orbits of period n-1. This is a map of the two-torus T^2 into itself. If we regard T^2 as R^2 / Z^2, the action of this map on a two vector in R^2 is multiplication by the unit-determinant matrix A = [2, 1;1, 1], with the vector components taken modulo one. As such, an explicit formula for the n-th entry of this sequence is -det(I-A^n). - _Bruce Boghosian_, Apr 26 2009

%C 7*a(n) gives the total number of vertices in a heptagonal hyperbolic lattice {7,3} with n total levels, in which an open heptagon is centered at the origin. - _Robert M. Ziff_, Apr 10 2011

%C The sequence is the case P1 = 5, P2 = 6, Q = 1 of the 3 parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - _Peter Bala_, Apr 03 2014

%C Determinants of the spiral knots S(3,k,(1,-1)). a(k) = det(S(3,k,(1,-1))). These knots are also the weaving knots W(k,3) and the Turk's Head Links THK(3,k). - _Ryan Stees_, Dec 14 2014

%C Even-indexed Fibonacci numbers (1, 3, 8, 21, ...) convolved with (1, 2, 2, 2, 2, ...). - _Gary W. Adamson_, Aug 09 2016

%C a(n) is the number of ways to tile a bracelet of length n with 1-color square, 2-color dominos, 3-color trominos, etc. - _Yu Xiao_, May 23 2020

%C a(n) is the number of face-labeled unfoldings of a pyramid whose base is a simple n-gon. Cf. A103536. - _Rick Mabry_, Apr 17 2023

%D I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, (p. 193, Problem 3.3.40 (a)).

%D N. Hartsfield and G. Ringel, Pearls in Graph Theory, p. 102. Academic Press: 1990.

%D B. Hasselblatt and A. Katok, "Introduction to the Modern Theory of Dynamical Systems," Cambridge University Press, 1997.

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

%H G. C. Greubel, <a href="/A004146/b004146.txt">Table of n, a(n) for n = 0..2375</a> (terms 0..200 from T. D. Noe)

%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 S. Barbero, U. Cerruti, and N. Murru, <a href="http://www.seminariomatematico.polito.it/rendiconti/78-1/BarberoCerrutiMurru.pdf">On polynomial solutions of the Diophantine equation (x + y - 1)^2 = wxy</a>, Rendiconti Sem. Mat. Univ. Pol. Torino (2020) Vol. 78, No. 1, 5-12.

%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 pp. 25, 29.

%H N. Brothers, S. Evans, L. Taalman, L. Van Wyk, D. Witczak, and C. Yarnall, <a href="http://projecteuclid.org/euclid.mjms/1312232716">Spiral knots</a>, Missouri J. of Math. Sci., 22 (2010).

%H M. DeLong, M. Russell, and J. Schrock, <a href="http://dx.doi.org/10.2140/involve.2015.8.361">Colorability and determinants of T(m,n,r,s) twisted torus knots for n equiv. +/-1(mod m)</a>, Involve, Vol. 8 (2015), No. 3, 361-384.

%H N. Dowdall, T. Mattman, K. Meek, and P. Solis, <a href="http://arxiv.org/abs/0811.0044">On the Harary-Kauffman conjecture and turk's head knots</a>, arxiv 0811.0044 [math.GT], 2008.

%H A. Flatters, <a href="http://www.arXiv.org/abs/0708.2190">Primitive divisors of some Lehmer-Pierce sequences</a>, arXiv:0708.2190 [math.NT], 2007.

%H Hang Gu and Robert M. Ziff, <a href="http://arxiv.org/abs/1111.5626">Crossing on hyperbolic lattices</a>, arXiv:1111.5626 [cond-mat.dis-nn], 2011 (see footnote 32).

%H Seong Ju Kim, R. Stees, and L. 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), #16.1.4.

%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 B. R. Myers, <a href="http://dx.doi.org/10.1109/TCT.1971.1083273">Number of spanning trees in a wheel</a>, IEEE Trans. Circuit Theory, 18 (1971), 280-282.

%H B. R. Myers, <a href="/A004146/a004146.pdf">Number of spanning trees in a wheel</a>, IEEE Trans. Circuit Theory, 18 (1971), 280-282. [Annotated scanned copy. See the last two pages of the scan, the first few pages are of a different article]

%H L. Oesper, <a href="http://educ.jmu.edu/~taalmala/OJUPKT/layla_thesis.pdf">p-Colorings of Weaving Knots</a>, Undergraduate Thesis, Pomona College, 2005.

%H Y. Puri and T. Ward, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL4/WARD/short.html">Arithmetic and growth of periodic orbits</a>, J. Integer Seqs., Vol. 4 (2001), #01.2.1.

%H Franck Ramaharo, <a href="https://arxiv.org/abs/1807.05256">A one-variable bracket polynomial for some Turk's head knots</a>, arXiv:1807.05256 [math.CO], 2018.

%H Mohammed A. Raouf, Fazirulhisyam Hashim, Jiun Terng Liew, and Kamal Ali Alezabi, <a href="https://doi.org/10.1371/journal.pone.0237386">Pseudorandom sequence contention algorithm for IEEE 802.11ah based internet of things network</a>, PLoS ONE (2020) Vol. 15, No. 8, e0237386.

%H K. R. Rebman, <a href="http://www.fq.math.ca/Scanned/13-1/rebman.pdf">The sequence: 1 5 16 45 121 320 ... in combinatorics</a>, Fib. Quart., 13 (1975), 51-55.

%H Harry Richman, Farbod Shokrieh, and Chenxi Wu, <a href="https://arxiv.org/abs/2308.03859">Counting two-forests and random cut size via potential theory</a>, arXiv:2308.03859 [math.CO], 2023. See p. 18.

%H Benoit Rittaud and Laurent Vivier, <a href="http://arxiv.org/abs/1108.3618">Circular words and applications</a>, arXiv:1108.3618 [cs.FL], 2011.

%H Benoit Rittaud and Laurent Vivier, <a href="https://hal.archives-ouvertes.fr/hal-00566314v1">Circular words and three applications: factors of the Fibonacci word, F -adic numbers, and the sequence 1, 5, 16, 45, 121, 320, ... </a>, HAL Id: hal-00566314.

%H Benoit Rittaud and Laurent Vivier, <a href="http://dx.doi.org/10.7169/facm/2012.47.2.6">Circular words and three applications: factors of the Fibonacci word, F -adic numbers, and the sequence 1, 5, 16, 45, 121, 320, ... </a>, Functiones et Approximatio Commentarii Mathematici, Volume 47, Number 2 (2012), 207-231.

%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/HelmGraph.html">Helm Graph</a>

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

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

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ArnoldsCatMap.html">Arnold's cat map</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Arnold&#39;s_cat_map">Arnold's cat map</a>

%H H. C. Williams and R. K. Guy, <a href="http://dx.doi.org/10.1142/S1793042111004587">Some fourth-order linear divisibility sequences</a>, Intl. J. Number Theory 7 (5) (2011) 1255-1277.

%H H. C. Williams and R. K. Guy, <a href="http://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_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,-4,1).

%F a(n) = A005248(n) - 2.

%F a(n+1) = 3*a(n) - a(n-1) + 2.

%F G.f.: x*(1+x)/(1-4*x+4*x^2-x^3) = x*(1+x)/((1-x)*(1-3*x+x^2)).

%F a(n) = 2*(T(n, 3/2)-1) with Chebyshev's polynomials T(n, x) of the first kind. See their coefficient triangle A053120.

%F a(n) = 4*a(n-1) - 4*a(n-2) + a(n-3), n>=3, a(0)=0, a(1)=1, a(2)=5.

%F a(n) = 2*T(n, 3/2) - 2, with twice the Chebyshev polynomials of the first kind, 2*T(n, x=3/2) = A005248(n).

%F a(n) = b(n) + b(n-1), n>=1, with b(n):=A027941(n-1), n>=1, b(-1):=0, the partial sums of S(n, 3) = U(n, 3/2) = A001906(n+1), with S(n, x) = U(n, x/2) Chebyshev's polynomials of the second kind.

%F a(2n) = A000204(2n)^2 - 4 = 5*A000045(2n)^2; a(2n+1) = A000204(2n+1)^2. - _David W. Wilson_, Jan 10 2006

%F a(n) = ((3+sqrt(5))/2)^n + ((3-sqrt(5))/2)^n - 2. - Felix Goldberg (felixg(AT)tx.technion.ac.il), Jun 09 2001

%F a(n) = b(n-1) + b(n-2), n>=1, with b(n):=A027941(n), b(-1):=0, partial sums of S(n, 3) = U(n, 3/2) = A001906(n+1), Chebyshev's polynomials of the second kind.

%F a(n) = n*Sum_{k=1..n} binomial(n+k-1,2*k-1)/k, n > 0. - _Vladimir Kruchinin_, Sep 03 2010

%F a(n) = floor(tau^(2*n)*(tau^(2*n) - floor(tau^(2*n)))), where tau = (1+sqrt(5))/2. - _L. Edson Jeffery_, Aug 26 2013

%F From _Peter Bala_, Apr 03 2014: (Start)

%F a(n) = U(n-1,sqrt(5)/2)^2, for n >= 1, where U(n,x) denotes the Chebyshev polynomial of the second kind.

%F a(n) = the bottom left entry of the 2 X 2 matrix T(n, M), where M is the 2 X 2 matrix [0, -3/2; 1, 5/2] and T(n,x) denotes the Chebyshev polynomial of the first kind.

%F See the remarks in A100047 for the general connection between Chebyshev polynomials of the first kind and 4th-order linear divisibility sequences. (End)

%F a(k) = det(S(3,k,(1,-1))) = b(k)^2, where b(1)=1, b(2)=sqrt(5), b(k)=sqrt(5)*b(k-1) - b(k-2) = b(2)*b(k-1) - b(k-2). - _Ryan Stees_, Dec 14 2014

%F exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + Sum_{n >= 1} Fibonacci(2*n)*x^n. Cf. A001350. - _Peter Bala_, Mar 19 2015

%F E.g.f.: exp(phi^2*x) + exp(x/phi^2) - 2*exp(x), where phi = (1 + sqrt(5))/2. - _G. C. Greubel_, Aug 24 2015

%F a(n) = a(-n) for all n in Z. - _Michael Somos_, Aug 27 2015

%F From _Peter Bala_, Jun 03 2016: (Start)

%F a(n) = Lucas(2*n) - Lucas(0*n);

%F a(n)^2 = Lucas(4*n) - 3*Lucas(2*n) + 3*Lucas(0*n) - Lucas(-2*n);

%F a(n)^3 = Lucas(6*n) - 5*Lucas(4*n) + 10*Lucas(2*n) - 10*Lucas(0*n) + 5*Lucas(-2*n) - Lucas(-4*n) and so on (follows from Binet's formula for Lucas(2*n) and the algebraic identity (x + 1/x - 2)^m = f(x) + f(1/x) where f(x) = (x - 1)^(2*m - 1)/x^(m-1) ). (End)

%F Limit_{n->infinity} a(n+1)/a(n) = (3 + sqrt(5))/2 = A104457. - _Ilya Gutkovskiy_, Jun 03 2016

%F a(n) = (phi^n - phi^(-n))^2, where phi = A001622 = (1 + sqrt(5))/2. - _Diego Rattaggi_, Jun 10 2020

%F a(n) = 4*sinh(n*A002390)^2, where A002390 = arcsinh(1/2). - _Gleb Koloskov_, Sep 18 2021

%F a(n) = 5*F(n)^2 = L(n)^2 - 4 if n even and a(n) = L(n)^2 = 5*F(n)^2 - 4 if n odd. - _Michael Somos_, Feb 10 2023

%e For k=3, b(3) = sqrt(5)*b(2) - b(1) = 5 - 1 = 4, so det(S(3,3,(1,-1))) = 4^2 = 16.

%e G.f. = x + 5*x^2 + 16*x^3 + 45*x^4 + 121*x^5 + 320*x^4 + 841*x^5 + ... - _Michael Somos_, Feb 10 2023

%t Table[LucasL[2*n] - 2, {n, 0, 20}]

%t (* Second program: *)

%t LinearRecurrence[{4, -4, 1}, {0, 1, 5}, 30] (* _Jean-François Alcover_, Jan 08 2019 *)

%o (PARI) a(n) = { we = quadgen(5);((1+we)^n) + ((2-we)^n) - 2;} /* _Michel Marcus_, Aug 18 2012 */

%o (Magma) [Lucas(n)-2: n in [0..60 by 2]]; // _Vincenzo Librandi_, Mar 20 2015

%Y This is the r=5 member of the family S_r(n) defined in A092184.

%Y Cf. A005248. Partial sums of A002878. Pairwise sums of A027941. Bisection of A074392.

%Y Sequence A032170, the Möbius transform of this sequence, is then the number of prime periodic orbits of Arnold's cat map. - _Bruce Boghosian_, Apr 26 2009

%Y Cf. A100047, A001350.

%Y Cf. also A158039, A158040, A158041, A158042, A158043, A158044.

%Y Cf. A103536 for the number of geometrically distinct edge-unfoldings of the regular pyramid. - _Rick Mabry_, Apr 17 2023

%K nonn,easy

%O 0,3

%A _N. J. A. Sloane_

%E Correction to formula from Nephi Noble (nephi(AT)math.byu.edu), Apr 09 2002

%E Chebyshev comments from _Wolfdieter Lang_, Sep 10 2004