A006235 Complexity of doubled cycle (regarding case n = 2 as a multigraph).
1, 12, 75, 384, 1805, 8100, 35287, 150528, 632025, 2620860, 10759331, 43804800, 177105253, 711809364, 2846259375, 11330543616, 44929049777, 177540878700, 699402223099, 2747583822720, 10766828545725, 42095796462852, 164244726238343, 639620518118400, 2486558615814025
Offset: 1
Examples
For k=3, b(3)=sqrt(6)b(2)-b(1)=6-1=5, so det(S(4,3,(1,-1,1)))=3*5^2=75.
References
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Links
- T. D. Noe, Table of n, a(n) for n = 1..200
- Zbigniew R. Bogdanowicz, The number of spanning trees in a superprism, Discrete Math. Lett. 13 (2024) 66-73. See p. 66.
- 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.
- N. Dowdall, T. Mattman, K. Meek, and P. Solis, On the Harary-Kauffman conjecture and turk's head knots, arxiv 0811.0044 [math.GT], 2008.
- A. A. Jagers, A note on the number of spanning trees in a prism graph, Int. J. Comput. Math., Vol. 24, 1988 (Issue 2), pp. 151-154.
- Seong Ju Kim, R. Stees, and L. Taalman, Sequences of Spiral Knot Determinants, Journal of Integer Sequences, Vol. 19 (2016), #16.1.4.
- D. E. Knuth, Letter to N. J. A. Sloane, Oct. 1994
- Germain Kreweras, Complexité et circuits Eulériens dans les sommes tensorielles de graphes, J. Combin. Theory, B 24 (1978), 202-212.
- L. Oesper, p-Colorings of Weaving Knots, Undergraduate Thesis, Pomona College, 2005.
- Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
- Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
- Ryan Stees, Sequences of Spiral Knot Determinants, Senior Honors Projects, Paper 84, James Madison Univ., May 2016.
- Eric Weisstein's World of Mathematics, Dipyramidal Graph
- Eric Weisstein's World of Mathematics, Prism Graph
- Eric Weisstein's World of Mathematics, Spanning Tree
- Eric Weisstein's World of Mathematics, Web Graph
- Index entries for linear recurrences with constant coefficients, signature (10,-35,52,-35,10,-1).
Crossrefs
Programs
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Maple
A006235:=(1+2*z-10*z**2+2*z**3+z**4)/(z-1)**2/(z**2-4*z+1)**2; # conjectured (correctly) by Simon Plouffe in his 1992 dissertation
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Mathematica
LinearRecurrence[{10, -35, 52, -35, 10, -1}, {0, 1, 12, 75, 384, 1805}, 20] Table[1/2 (-2 + (2 - Sqrt[3])^n + (2 + Sqrt[3])^n) n, {n, 0, 20}] // Expand Table[n (ChebyshevT[n, 2] - 1), {n, 20}] (* Eric W. Weisstein, Mar 30 2017 *) -
PARI
a(n)=if(n<0,0,polcoeff(x*(1+2*x-10*x^2+2*x^3+x^4)/((1-x)*(1-4*x+x^2))^2+x*O(x^n),n))
Formula
a(n) = (1/2)*n*(-2 + (2 - sqrt(3))^n + (2 + Sqrt(3))^n) (Kreweras). - Eric W. Weisstein, Jul 15 2011
G.f.: x*(1+2*x-10*x^2+2*x^3+x^4)/((1-x)*(1-4*x+x^2))^2.
a(n) = 10*a(n-1)-35*a(n-2)+52*a(n-3)-35*a(n-4)+10*a(n-5)-a(n-6), n>5.
a(n) = (n/2)*A129743(n). - Woong Kook and Seung Kyoon Shin (andrewk(AT)math.uri.edu), Jan 13 2009
a(k) = det(S(4,k,(1,-1,1))) = k*b(k)^2, where b(1)=1, b(2)=sqrt(6), b(k)=sqrt(6)*b(k-1) - b(k-2) = b(2)*b(k-1) - b(k-2). - Ryan Stees, Dec 14 2014
a(n) = n*(A001075(n) - 1). - Eric W. Weisstein, Mar 30 2017
E.g.f.: exp(x)*x*(exp(x)*(2*cosh(sqrt(3)*x) + sqrt(3)*sinh(sqrt(3)*x)) - 1). - Stefano Spezia, May 05 2024
Extensions
More terms from Michael Somos, Jul 19 2002
Minor edits by N. J. A. Sloane, May 27 2012
Comments