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A004146 Alternate Lucas numbers - 2.
(Formerly M3867)
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 (list; graph; refs; listen; history; text; internal format)



This is the r=5 member in the r-family 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 n-helm graph. - _Eric 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)= (n-1)!

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

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+1)th entry of this sequence is det(I-A^n). - Bruce Boghosian (bruce.boghosian(AT)tufts.edu), 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 4-th order 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


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), To appear in Involve.

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. [From Bruce Boghosian (bruce.boghosian(AT)tufts.edu), Apr 26 2009]

L. Oesper, p-Colorings of Weaving Knots, Undergraduate Thesis, Pomona College, 2005.

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,. . . . To appear in Functiones et Approximatio Commentarii Mathematici.

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


T. D. Noe, Table of n, a(n) for n=0..200

N. Brothers, S. Evans, L. Taalman, L. Van Wyk, D. Witczak, and C. Yarnall, Spiral knots, Missouri J. of Math. Sci., 22 (2010).

N. Dowdall, T. Mattman, K. Meek, and P. Solis, On the Harary-Kauffman conjecture and turk's head knots, arxiv 0811.0044v1.

A. Flatters, Primitive divisors of some Lehmer-Pierce sequences

Hang Gu and Robert M. Ziff, Crossing on hyperbolic lattices, arXiv preprint arXiv:1111.5626, 2011 (see footnote 32).

B. R. Myers, Number of spanning trees in a wheel, IEEE Trans. Circuit Theory, 18 (1971), 280-282.

Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.

K. R. Rebman, The sequence: 1 5 16 45 121 320 ... in combinatorics, Fib. Quart., 13 (1975), 51-55.

Benoit Rittaud and Laurent Vivier, Circular words and applications, arXiv preprint arXiv:1108.3618, 2011

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 fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.

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 to sequences with linear recurrences with constant coefficients, signature (4,-4,1).


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

G.f.: x*(1+x)/(1-4*x+4*x^2-x^3) = x*(1+x)/((1-x)*(1-3*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(n-1)-4*a(n-2)+a(n-3), 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(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.

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

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.

a(n) = n*sum(binomial(n+k-1,2*k-1)/k,k,1,n), 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(n-1,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 2X2 matrix T(n, M), where M is the 2X2 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 4-th order 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(k-1) - b(k-2) = b(2)*b(k-1) - b(k-2). - Ryan Stees, Dec 14 2014


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


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


(PARI) a(n) = { we = quadgen(5); ((1+we)^n) + ((2-we)^n) - 2; } /* Michel Marcus, Aug 18 2012 */


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

Equals A005248 - 2. Partial sums of A002878. Pairwise sums of A027941. Bisection of A074392.

Sequence A032170, the Moebius transform of this sequence, is then the number of prime periodic orbits of Arnold's cat map. - Bruce Boghosian (bruce.boghosian(AT)tufts.edu), Apr 26 2009

Cf. A100047.

Sequence in context: A185003 A189390 A099327 * A071101 A110580 A055552

Adjacent sequences:  A004143 A004144 A004145 * A004147 A004148 A004149




N. J. A. Sloane


More terms from Larry Reeves (larryr(AT)acm.org), Jun 11 2001. Correction to formula from Nephi Noble (nephi(AT)math.byu.edu), Apr 09 2002

Chebyshev comments from Wolfdieter Lang, Sep 10 2004



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Last modified February 28 22:04 EST 2015. Contains 255096 sequences.