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A004146
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Alternate Lucas numbers - 2.
(Formerly M3867)
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37
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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
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0,3
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COMMENTS
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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
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
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)!.
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 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
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 4th-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
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 1-color square, 2-color dominos, 3-color trominos, etc. - Yu Xiao, May 23 2020
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
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REFERENCES
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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).
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LINKS
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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, 361-384.
B. R. Myers, Number of spanning trees in a wheel, 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]
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: hal-00566314.
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), 207-231.
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FORMULA
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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 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(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) = floor(tau^(2*n)*(tau^(2*n) - floor(tau^(2*n)))), where tau = (1+sqrt(5))/2. - L. Edson Jeffery, Aug 26 2013
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 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 4th-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
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) = 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^(m-1) ). (End)
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
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EXAMPLE
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For k=3, b(3) = sqrt(5)*b(2) - b(1) = 5 - 1 = 4, so det(S(3,3,(1,-1))) = 4^2 = 16.
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
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MATHEMATICA
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Table[LucasL[2*n] - 2, {n, 0, 20}]
(* Second program: *)
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PROG
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(PARI) a(n) = { we = quadgen(5); ((1+we)^n) + ((2-we)^n) - 2; } /* Michel Marcus, Aug 18 2012 */
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CROSSREFS
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This is the r=5 member of the family S_r(n) defined in A092184.
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. A103536 for the number of geometrically distinct edge-unfoldings of the regular pyramid. - Rick Mabry, Apr 17 2023
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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Correction to formula from Nephi Noble (nephi(AT)math.byu.edu), Apr 09 2002
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STATUS
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approved
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