%I A004146 M3867
%S A004146 0,1,5,16,45,121,320,841,2205,5776,15125,39601,103680,271441,710645,
%T A004146 1860496,4870845,12752041,33385280,87403801,228826125,599074576,
%U A004146 1568397605,4106118241,10749957120,28143753121,73681302245
%N A004146 Alternate Lucas numbers - 2.
%C A004146 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 A004146 Number of spanning trees of the wheel W_n on n+1 vertices. - Emeric Deutsch
(deutsch(AT)duke.poly.edu), Mar 27 2005
%C A004146 a(n) is the smallest number requiring n terms when expressed as a sum
of lucas numbers (A000204). - David W. Wilson (davidwwilson(AT)comcast.net),
Jan 10 2006
%C A004146 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 A004146 Contribution from Giorgio Balzarotti (greenblue(AT)tiscali.it), Mar 11
2009: (Start)
%C A004146 Determinant of power series of gamma matrix with determinant 1!
%C A004146 a(n) = Determinant( A+A^2+ A^3+ A^4+ A^5+... A^n)
%C A004146 where A is the submatrix A(1..2,1..2)= of the matrix with factorial determinant
%C A004146 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 A004146 [1,2,3,4,5,1,...],[1,2,3,4,5,6,...],...] note: Determinant A(1..n,1..n)=
(n-1)!
%C A004146 See A158039, A158040, A158041, A158042, A158043, A158044, for sequences
of
%C A004146 matrix 2!,3!,.. (End)
%C A004146 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]
%D A004146 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences,
Academic Press, 1995 (includes this sequence).
%D A004146 I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y.,
1983,(p.193, Problem 3.3.40 (a)).
%D A004146 N. Hartsfield and G. Ringel, Pearls in Graph Theory, p. 102. Academic
Press: 1990.
%D A004146 B. R. Myers, Number of spanning trees in a wheel, IEEE Trans. Circuit
Theory, 18 (1971), 280-282.
%D A004146 K. R. Rebman, The sequence: 1 5 16 45 121 320 ... in combinatorics, Fib.
Quart., 13 (1975), 51-55.
%D A004146 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]
%H A004146 T. D. Noe, <a href="b004146.txt">Table of n, a(n) for n=0..200</a>
%H A004146 Y. Puri and T. Ward, <a href="http://www.cs.uwaterloo.ca/journals/JIS/
index.html">Arithmetic and growth of periodic orbits</a>, J. Integer
Seqs., Vol. 4 (2001), #01.2.1.
%H A004146 <a href="Sindx_Ch.html#Cheby">Index entries for sequences related to
Chebyshev polynomials.</a>
%H A004146 Author?, <a href="http://www.arXiv.org/pdf/0708.2190">Title?</a>
%H A004146 <A href="http://en.wikipedia.org/wiki/Arnold's_cat_map">Wikipedia entry
for Arnold's cat map</A> [From Bruce Boghosian (bruce.boghosian(AT)tufts.edu),
Apr 26 2009]
%H A004146 <A href="http://mathworld.wolfram.com/ArnoldsCatMap.html">Wolfram MathWorld
entry for Arnold's cat map</A> [From Bruce Boghosian (bruce.boghosian(AT)tufts.edu),
Apr 26 2009]
%F A004146 a(n+1)=3*a(n)-a(n-1)+2.
%F A004146 G.f.: x*(1+x)/(1-4*x+4*x^2-x^3) = x*(1+x)/((1-x)*(1-3*x+x^2))
%F A004146 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 A004146 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 A004146 G.f.: x*(1+x)/((1-x)*(1-3*x+x^2))=x*(1+x)/(1-4*x+4*x^2-x^3).
%F A004146 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).
%F A004146 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 A004146 a(2n) = A000204(2n)^2-4 = 5*A000045(2n)^2; a(2n+1) = A000204(2n+1)^2
- David W. Wilson (davidwwilson(AT)comcast.net), Jan 10 2006
%F A004146 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 A004146 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.
%Y A004146 This is the r=5 member of the family S_r(n) defined in A092184.
%Y A004146 Equals A005248 - 2. Partial sums of A002878. Pairwise sums of A027941.
Bisection of A074392.
%Y A004146 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]
%Y A004146 Sequence in context: A053220 A048777 A099327 this_sequence A071101 A110580
A055552
%Y A004146 Adjacent sequences: A004143 A004144 A004145 this_sequence A004147 A004148
A004149
%K A004146 nonn,easy
%O A004146 0,3
%A A004146 N. J. A. Sloane (njas(AT)research.att.com).
%E A004146 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.
%E A004146 Chebyshev comments from W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de),
Sep 10 2004
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