

A004146


Alternate Lucas numbers  2.
(Formerly M3867)


31



0, 1, 5, 16, 45, 121, 320, 841, 2205, 5776, 15125, 39601, 103680, 271441, 710645, 1860496, 4870845, 12752041, 33385280, 87403801, 228826125, 599074576, 1568397605, 4106118241, 10749957120, 28143753121, 73681302245
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OFFSET

0,3


COMMENTS

This is the r=5 member in the rfamily of sequences S_r(n) defined in A092184 where more information can be found.
Number of spanning trees of the wheel W_n on n+1 vertices.  Emeric Deutsch, Mar 27 2005
Also number of spanning trees of the nhelm graph.  Eric W. Weisstein, Jul 16 2011
a(n) is the smallest number requiring n terms when expressed as a sum of Lucas numbers (A000204).  David W. Wilson, Jan 10 2006
This sequence has a primitive prime divisor for all terms beyond the twelfth.  Anthony Flatters (Anthony.Flatters(AT)uea.ac.uk), Aug 17 2007
From Giorgio Balzarotti, Mar 11 2009: (Start)
Determinant of power series of gamma matrix with determinant 1:
a(n) = Determinant(A + A^2 + A^3 + A^4 + A^5 + ... + A^n)
where A is the submatrix A(1..2,1..2) of the matrix with factorial determinant
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,...],
[1,2,3,4,5,1,...],[1,2,3,4,5,6,...],...]. Note: Determinant A(1..n,1..n)= (n1)!.
See A158039, A158040, A158041, A158042, A158043, A158044, for sequences of matrix 2!,3!,... (End)
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
a(n) is also the number of points of Arnold's "cat map" that are on orbits of period n1. This is a map of the twotorus 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 unitdeterminant matrix A = [2, 1;1, 1], with the vector components taken modulo one. As such, an explicit formula for the nth entry of this sequence is det(IA^n).  Bruce Boghosian, Apr 26 2009
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
The sequence is the case P1 = 5, P2 = 6, Q = 1 of the 3 parameter family of 4thorder linear divisibility sequences found by Williams and Guy.  Peter Bala, Apr 03 2014
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
Even indexed Fibonacci numbers (1, 3, 8, 21, ...) convolved with (1, 2, 2, 2, 2, ...).  Gary W. Adamson, Aug 09 2016
a(n) is the number of ways to tile a bracelet of length n with 1color square, 2color dominos, 3color trominos, etc.  Yu Xiao, May 23 2020


REFERENCES

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, (p. 193, Problem 3.3.40 (a)).
N. Hartsfield and G. Ringel, Pearls in Graph Theory, p. 102. Academic Press: 1990.
B. Hasselblatt and A. Katok, "Introduction to the Modern Theory of Dynamical Systems," Cambridge University Press, 1997.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS

G. C. Greubel, Table of n, a(n) for n = 0..2375 (terms 0..200 from T. D. Noe)
Marco Abrate, Stefano Barbero, Umberto Cerruti, and Nadir Murru, Polynomial sequences on quadratic curves, Integers, Vol. 15, 2015, #A38.
S. Barbero, U. Cerruti, and N. Murru, On polynomial solutions of the Diophantine equation (x + y  1)^2 = wxy, Rendiconti Sem. Mat. Univ. Pol. Torino (2020) Vol. 78, No. 1, 512.
N. Brothers, S. Evans, L. Taalman, L. Van Wyk, D. Witczak, and C. Yarnall, Spiral knots, Missouri J. of Math. Sci., 22 (2010).
M. DeLong, M. Russell, and J. Schrock, Colorability and determinants of T(m,n,r,s) twisted torus knots for n equiv. +/1(mod m), Involve, Vol. 8 (2015), No. 3, 361384.
N. Dowdall, T. Mattman, K. Meek, and P. Solis, On the HararyKauffman conjecture and turk's head knots, arxiv 0811.0044 [math.GT], 2008.
A. Flatters, Primitive divisors of some LehmerPierce sequences, arXiv:0708.2190 [math.NT], 2007.
Hang Gu and Robert M. Ziff, Crossing on hyperbolic lattices, arXiv:1111.5626 [condmat.disnn], 2011 (see footnote 32).
Seong Ju Kim, R. Stees, and L. Taalman, Sequences of Spiral Knot Determinants, Journal of Integer Sequences, Vol. 19 (2016), #16.1.4.
IoanaClaudia Lazăr, Lucas sequences in tuniform simplicial complexes, arXiv:1904.06555 [math.GR], 2019.
B. R. Myers, Number of spanning trees in a wheel, IEEE Trans. Circuit Theory, 18 (1971), 280282.
B. R. Myers, Number of spanning trees in a wheel, IEEE Trans. Circuit Theory, 18 (1971), 280282. [Annotated scanned copy. See the last two pages of the scan, the first few pages are of a different article]
L. Oesper, pColorings of Weaving Knots, Undergraduate Thesis, Pomona College, 2005.
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Franck Ramaharo, A onevariable bracket polynomial for some Turk's head knots, arXiv:1807.05256 [math.CO], 2018.
Mohammed A. Raouf, Fazirulhisyam Hashim, Jiun Terng Liew, and Kamal Ali Alezabi, Pseudorandom sequence contention algorithm for IEEE 802.11ah based internet of things network, PLoS ONE (2020) Vol. 15, No. 8, e0237386.
K. R. Rebman, The sequence: 1 5 16 45 121 320 ... in combinatorics, Fib. Quart., 13 (1975), 5155.
Benoit Rittaud and Laurent Vivier, Circular words and applications, arXiv:1108.3618 [cs.FL], 2011.
Benoit Rittaud and Laurent Vivier, Circular words and three applications: factors of the Fibonacci word, F adic numbers, and the sequence 1, 5, 16, 45, 121, 320, ... , HAL Id: hal00566314.
Benoit Rittaud and Laurent Vivier, Circular words and three applications: factors of the Fibonacci word, F adic numbers, and the sequence 1, 5, 16, 45, 121, 320, ... , Functiones et Approximatio Commentarii Mathematici, Volume 47, Number 2 (2012), 207231.
Ryan Stees, Sequences of Spiral Knot Determinants, Senior Honors Projects. Paper 84. James Madison Univ., May 2016.
Eric Weisstein's World of Mathematics, Helm Graph
Eric Weisstein's World of Mathematics, Spanning Tree
Eric Weisstein's World of Mathematics, Wheel Graph
Eric Weisstein's World of Mathematics, Arnold's cat map
Wikipedia, Arnold's cat map
H. C. Williams and R. K. Guy, Some fourthorder linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 12551277.
H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences Integers, Volume 12A (2012) The John Selfridge Memorial Volume
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (4,4,1).


FORMULA

a(n) = A005248(n)  2.
a(n+1) = 3*a(n)  a(n1) + 2.
G.f.: x*(1+x)/(14*x+4*x^2x^3) = x*(1+x)/((1x)*(13*x+x^2)).
a(n) = 2*(T(n, 3/2)1) with Chebyshev's polynomials T(n, x) of the first kind. See their coefficient triangle A053120.
a(n) = 4*a(n1)  4*a(n2) + a(n3), n>=3, a(0)=0, a(1)=1, a(2)=5.
a(n) = 2*T(n, 3/2)  2, with twice the Chebyshev's polynomials of the first kind, 2*T(n, x=3/2) = A005248(n).
a(n) = b(n) + b(n1), n>=1, with b(n):=A027941(n1), 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.
a(2n) = A000204(2n)^2  4 = 5*A000045(2n)^2; a(2n+1) = A000204(2n+1)^2.  David W. Wilson, Jan 10 2006
a(n) = ((3+sqrt(5))/2)^n + ((3sqrt(5))/2)^n  2.  Felix Goldberg (felixg(AT)tx.technion.ac.il), Jun 09 2001
a(n) = b(n1) + b(n2), 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.
a(n) = n*Sum_{k=1..n} binomial(n+k1,2*k1)/k, n > 0.  Vladimir Kruchinin, Sep 03 2010
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
From Peter Bala, Apr 03 2014: (Start)
a(n) = U(n1,sqrt(5)/2)^2, for n >= 1, where U(n,x) denotes the Chebyshev polynomial of the second kind.
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.
See the remarks in A100047 for the general connection between Chebyshev polynomials of the first kind and 4thorder linear divisibility sequences. (End)
a(k) = det(S(3,k,(1,1))) = b(k)^2, where b(1)=1, b(2)=sqrt(5), b(k)=sqrt(5)*b(k1)  b(k2) = b(2)*b(k1)  b(k2).  Ryan Stees, Dec 14 2014
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
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
a(n) = a(n) for all n in Z.  Michael Somos, Aug 27 2015
From Peter Bala, Jun 03 2016: (Start)
a(n) = Lucas(2*n)  Lucas(0*n);
a(n)^2 = Lucas(4*n)  3*Lucas(2*n) + 3*Lucas(0*n)  Lucas(2*n);
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^(m1) ). (End)
Lim_{n>infinity} a(n+1)/a(n) = (3 + sqrt(5))/2 = A104457.  Ilya Gutkovskiy, Jun 03 2016
a(n) = (phi^n  phi^(n))^2, where phi = A001622 = (1 + sqrt(5))/2.  Diego Rattaggi, Jun 10 2020
a(n) = 4*sinh(n*A002390)^2, where A002390 = arcsinh(1/2).  Gleb Koloskov, Sep 18 2021


EXAMPLE

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


MATHEMATICA

Table[LucasL[2*n]  2, {n, 0, 20}]
(* Second program: *)
LinearRecurrence[{4, 4, 1}, {0, 1, 5}, 30] (* JeanFrançois Alcover, Jan 08 2019 *)


PROG

(PARI) a(n) = { we = quadgen(5); ((1+we)^n) + ((2we)^n)  2; } /* Michel Marcus, Aug 18 2012 */
(MAGMA) [Lucas(n)2: n in [0..60 by 2]]; // Vincenzo Librandi, Mar 20 2015


CROSSREFS

This is the r=5 member of the family S_r(n) defined in A092184.
Cf. A005248. Partial sums of A002878. Pairwise sums of A027941. Bisection of A074392.
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
Cf. A100047, A001350.
Cf. also A158039, A158040, A158041, A158042, A158043, A158044.
Sequence in context: A185003 A189390 A099327 * A275126 A071101 A110580
Adjacent sequences: A004143 A004144 A004145 * A004147 A004148 A004149


KEYWORD

nonn,easy


AUTHOR

N. J. A. Sloane


EXTENSIONS

Correction to formula from Nephi Noble (nephi(AT)math.byu.edu), Apr 09 2002
Chebyshev comments from Wolfdieter Lang, Sep 10 2004


STATUS

approved



