Search: id:A004146 Results 1-1 of 1 results found. %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, Table of n, a(n) for n=0..200 %H A004146 Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1. %H A004146 Index entries for sequences related to Chebyshev polynomials. %H A004146 Author?, Title? %H A004146 Wikipedia entry for Arnold's cat map [From Bruce Boghosian (bruce.boghosian(AT)tufts.edu), Apr 26 2009] %H A004146 Wolfram MathWorld entry for Arnold's cat map [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 Search completed in 0.002 seconds