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 A003948 Expansion of (1+x)/(1-5*x). 64
 1, 6, 30, 150, 750, 3750, 18750, 93750, 468750, 2343750, 11718750, 58593750, 292968750, 1464843750, 7324218750, 36621093750, 183105468750, 915527343750, 4577636718750, 22888183593750, 114440917968750, 572204589843750, 2861022949218750, 14305114746093750 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Coordination sequence for infinite tree with valency 6. The n-th term of the coordination sequence of the infinite tree with valency 2m is the same as the number of reduced words of size n in the free group on m generators. In the five sequences A003946, A003948, A003950, A003952, A003954, m is 2, 3, 4, 5, 6. - Avi Peretz (njk(AT)netvision.net.il), Feb 23 2001 and Ola Veshta (olaveshta(AT)my-deja.com), Mar 30 2001 Hamiltonian path in S_4 X P_2n. For n>=1, a(n+1) is equal to the number of functions f:{1,2,...,n+1}->{1,2,3,4,5,6} such that for fixed, different x_1, x_2,...,x_n in {1,2,...,n+1} and fixed y_1, y_2,...,y_n in {1,2,3,4,5,6} we have f(x_i)<>y_i, (i=1..n). - Milan Janjic, May 10 2007 For n>=1, a(n) equals the numbers of words of length n over the alphabet {0..5} with no two adjacent letters identical. - Milan Janjic, Jan 31 2015 [Corrected by David Nacin, May 30 2017] a(n) equals the numbers of sequences of length n on {0,...,5} where no two adjacent terms differ by three. - David Nacin, May 30 2017 LINKS T. D. Noe, Table of n, a(n) for n = 0..200 F. Faase, Counting Hamiltonian cycles in product graphs INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 307 Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets A. M. Nemirovsky et al., Marriage of exact enumeration and 1/d expansion methods: lattice model of dilute polymers, J. Statist. Phys., 67 (1992), 1083-1108. Index to divisibility sequences Index entries for linear recurrences with constant coefficients, signature (5). Index entries for sequences related to trees FORMULA G.f.: (1+x)/(1-5*x). a(n) = Sum_{k=0..n} A029653(n, k)*x^k for x = 4. - Philippe Deléham, Jul 10 2005 The Hankel transform of this sequence is [1,-6,0,0,0,0,0,0,0,0,...]. - Philippe Deléham, Nov 21 2007 a(n) = 6*5^(n-1) for n>0, a(0)=1. - Vincenzo Librandi, Nov 18 2010 G.f.: 2/x - 5 - 8/(x*U(0)) where U(k)= 1 + 2/(3^k - 3^k/(2 + 1 - 12*x*3^k/(6*x*3^k + 1/U(k+1)))) ; (continued fraction, 4-step). - Sergei N. Gladkovskii, Oct 30 2012 E.g.f.: (6*exp(5*x) - 1)/5. - Ilya Gutkovskiy, Dec 10 2016 Sum_{n>=0} 1/a(n) = 29/24. - Bernard Schott, Oct 25 2021 MAPLE k := 6; if n = 0 then 1 else k*(k-1)^(n-1); fi; MATHEMATICA q = 6; Join[{a = 1}, Table[If[n != 0, a = q*a - a, a = q*a], {n, 0, 25}]] (* and *) Join[{1}, 6*5^Range[0, 25]] (* Vladimir Joseph Stephan Orlovsky, Jul 11 2011 *) Join[{1}, NestList[5#&, 6, 30]] (* Harvey P. Dale, Dec 31 2013 *) CoefficientList[Series[(1+x)/(1-5x), {x, 0, 30}], x] (* Michael De Vlieger, Dec 10 2016 *) PROG (PARI) Vec((1+x)/(1-5*x)+O(x^30)) \\ Charles R Greathouse IV, Nov 20 2012 (Magma) [1] cat [6*5^(n-1): n in [1..30]]; // G. C. Greubel, Sep 24 2019 (Sage) [1]+[6*5^(n-1) for n in (1..30)] # G. C. Greubel, Sep 24 2019 (GAP) Concatenation([1], List([1..30], n-> 6*5^(n-1) )); # G. C. Greubel, Sep 24 2019 CROSSREFS Cf. A003946, A003947, A003949, A003950, A003952, A003954, A029653. Sequence in context: A170591 A170639 A170687 * A105488 A252699 A054117 Adjacent sequences: A003945 A003946 A003947 * A003949 A003950 A003951 KEYWORD nonn,easy,nice,walk AUTHOR N. J. A. Sloane EXTENSIONS Definition corrected by Frans J. Faase, Feb 07 2009 Edited by N. J. A. Sloane, Dec 04 2009 STATUS approved

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Last modified April 21 13:26 EDT 2024. Contains 371870 sequences. (Running on oeis4.)