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A004146
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Alternate Lucas numbers - 2.
(Formerly M3867)
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13
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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
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
Contribution 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). [From 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, April 10, 2011
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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)).
Hang Gu and Robert M. Ziff, Crossing on hyperbolic lattices, Arxiv preprint arXiv:1111.5626, 2011 (see footnote 32).
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]
B. R. Myers, Number of spanning trees in a wheel, IEEE Trans. Circuit Theory, 18 (1971), 280-282.
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
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).
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..200
A. Flatters, Primitive divisors of some Lehmer-Pierce sequences
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Index entries for sequences related to Chebyshev polynomials.
Wikipedia, Arnold's cat map
Wolfram MathWorld entry for Arnold's cat map [From Bruce Boghosian (bruce.boghosian(AT)tufts.edu), Apr 26 2009]
Index to sequences with linear recurrences with constant coefficients, signature (4,-4,1).
Eric Weisstein's World of Mathematics, Helm Graph
Eric Weisstein's World of Mathematics, Spanning Tree
Eric Weisstein's World of Mathematics, Wheel Graph
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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'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. [From Vladimir Kruchinin, Sep 03 2010]
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MATHEMATICA
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Table[LucasL[2 n] - 2, {n, 0, 20}]
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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.
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. [From Bruce Boghosian (bruce.boghosian(AT)tufts.edu), Apr 26 2009]
Sequence in context: A185003 A189390 A099327 * A071101 A110580 A055552
Adjacent sequences: A004143 A004144 A004145 * A004147 A004148 A004149
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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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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STATUS
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approved
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