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URL: https://oeis.org/A006235

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A006235
Complexity of doubled cycle (regarding case n = 2 as a multigraph).
7
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
1,2
COMMENTS
In plain English, a(n) is the number of spanning trees of the n-prism graph Y_n. - Eric W. Weisstein, Jul 15 2011
Also the number of spanning trees of the n-web graph. - Eric W. Weisstein, Jul 15 2011
Also the number of spanning trees of the n-dipyramidal graph. - Eric W. Weisstein, Jun 14 2018
Determinants of the spiral knots S(4,k,(1,-1,1)). a(k) = det(S(4,k,(1,-1,1))). These knots are also the weaving knots W(k,4) and the Turk's Head Links THK(4,k). - Ryan Stees, Dec 14 2014
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
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).
Matt DeLong, Matthew Russell, and Jonathan 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.
Nicholas E. Dowdall, Thomas W. Mattman, Kevin Meek, and Pablo R. 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.
Germain Kreweras, Complexité et circuits Eulériens dans les sommes tensorielles de graphes, J. Combin. Theory, B 24 (1978), 202-212.
Layla 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.
Franck Ramaharo, A note on the recursive computation of the bracket polynomial for closed 4-tangles, arXiv:2508.17904 [math.GT], 2025. See p. 8.
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
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
EXAMPLE
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.
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
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 *)
PROG
(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))
CROSSREFS
Cf. A006237. Apart from a(2) coincides with A072373. A row or column of A173958.
Sequence in context: A292532 A053310 A368530 * A009642 A051104 A044199
KEYWORD
nonn,easy
EXTENSIONS
More terms from Michael Somos, Jul 19 2002
Minor edits by N. J. A. Sloane, May 27 2012
STATUS
approved