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%I M4849 #87 May 05 2024 20:01:26
%S 1,12,75,384,1805,8100,35287,150528,632025,2620860,10759331,43804800,
%T 177105253,711809364,2846259375,11330543616,44929049777,177540878700,
%U 699402223099,2747583822720,10766828545725,42095796462852,164244726238343,639620518118400,2486558615814025
%N Complexity of doubled cycle (regarding case n = 2 as a multigraph).
%C In plain English, a(n) is the number of spanning trees of the n-prism graph Y_n. - _Eric W. Weisstein_, Jul 15 2011
%C Also the number of spanning trees of the n-web graph. - _Eric W. Weisstein_, Jul 15 2011
%C Also the number of spanning trees of the n-dipyramidal graph. - _Eric W. Weisstein_, Jun 14 2018
%C 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
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A006235/b006235.txt">Table of n, a(n) for n = 1..200</a>
%H Zbigniew R. Bogdanowicz, <a href="https://www.dmlett.com/archive/v13/DML24_v13_pp66-73.pdf">The number of spanning trees in a superprism</a>, Discrete Math. Lett. 13 (2024) 66-73. See p. 66.
%H N. Brothers, S. Evans, L. Taalman, L. Van Wyk, D. Witczak, and C. Yarnall, <a href="http://projecteuclid.org/euclid.mjms/1312232716">Spiral knots</a>, Missouri J. of Math. Sci., 22 (2010).
%H M. DeLong, M. Russell, and J. Schrock, <a href="http://dx.doi.org/10.2140/involve.2015.8.361">Colorability and determinants of T(m,n,r,s) twisted torus knots for n equiv. +/-1(mod m)</a>, Involve, Vol. 8 (2015), No. 3, 361-384.
%H N. Dowdall, T. Mattman, K. Meek, and P. Solis, <a href="http://arxiv.org/abs/0811.0044">On the Harary-Kauffman conjecture and turk's head knots</a>, arxiv 0811.0044 [math.GT], 2008.
%H A. A. Jagers, <a href="http://dx.doi.org/10.1080/00207168808803639">A note on the number of spanning trees in a prism graph</a>, Int. J. Comput. Math., Vol. 24, 1988 (Issue 2), pp. 151-154.
%H Seong Ju Kim, R. Stees, and L. Taalman, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Stees/stees4.html">Sequences of Spiral Knot Determinants</a>, Journal of Integer Sequences, Vol. 19 (2016), #16.1.4.
%H D. E. Knuth, <a href="/A006235/a006235.pdf">Letter to N. J. A. Sloane, Oct. 1994</a>
%H Germain Kreweras, <a href="http://dx.doi.org/10.1016/0095-8956(78)90021-7">Complexité et circuits Eulériens dans les sommes tensorielles de graphes</a>, J. Combin. Theory, B 24 (1978), 202-212.
%H L. Oesper, <a href="http://educ.jmu.edu/~taalmala/OJUPKT/layla_thesis.pdf">p-Colorings of Weaving Knots</a>, Undergraduate Thesis, Pomona College, 2005.
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H Ryan Stees, <a href="https://commons.lib.jmu.edu/honors201019/84">Sequences of Spiral Knot Determinants</a>, Senior Honors Projects, Paper 84, James Madison Univ., May 2016.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DipyramidalGraph.html">Dipyramidal Graph</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrismGraph.html">Prism Graph</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SpanningTree.html">Spanning Tree</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/WebGraph.html">Web Graph</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (10,-35,52,-35,10,-1).
%F a(n) = (1/2)*n*(-2 + (2 - sqrt(3))^n + (2 + Sqrt(3))^n) (Kreweras). - _Eric W. Weisstein_, Jul 15 2011
%F G.f.: x*(1+2*x-10*x^2+2*x^3+x^4)/((1-x)*(1-4*x+x^2))^2.
%F 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.
%F a(n) = (n/2)*A129743(n). - Woong Kook and Seung Kyoon Shin (andrewk(AT)math.uri.edu), Jan 13 2009
%F 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
%F a(n) = n*(A001075(n) - 1). - _Eric W. Weisstein_, Mar 30 2017
%F 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
%e 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.
%p 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
%t LinearRecurrence[{10, -35, 52, -35, 10, -1}, {0, 1, 12, 75, 384, 1805}, 20]
%t Table[1/2 (-2 + (2 - Sqrt[3])^n + (2 + Sqrt[3])^n) n, {n, 0, 20}] // Expand
%t Table[n (ChebyshevT[n, 2] - 1), {n, 20}] (* _Eric W. Weisstein_, Mar 30 2017 *)
%o (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))
%Y Cf. A006237. Apart from a(2) coincides with A072373. A row or column of A173958.
%Y Cf. A001075, A129743.
%K nonn,easy
%O 1,2
%A _N. J. A. Sloane_
%E More terms from _Michael Somos_, Jul 19 2002
%E Minor edits by _N. J. A. Sloane_, May 27 2012
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