A198808 Number of closed paths of length n whose steps are 12th roots of unity, U_12(n).
1, 0, 12, 24, 396, 2160, 23160, 186480, 1845900, 17213280, 171575712, 1703560320, 17365421304, 178323713568, 1856554560432, 19487791106784, 206411964321420, 2201711191213248, 23642813637773616, 255355132936441824, 2772650461148938656
Offset: 0
Keywords
Links
- Robert Israel, Table of n, a(n) for n = 0..929
- V. Braun, P. Candelas, X. de la Ossa, Two One-Parameter Special Geometries, arXiv preprint arXiv:1512.08367 [hep-th], 2015.
- Gilbert Labelle and Annie Lacasse, Closed paths whose steps are roots of unity, in FPSAC 2011, Reykjavik, Iceland DMTCS proc. AO, 2011, 599-610.
Programs
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Maple
N:= 50: # to get a(0)..a(N) U6:= rectoproc({(36*n^2+180*n+216)*a6(n+1)+(24*n^2+144*n+216)*a6(n+2)+(n^2+7*n+12)*a6(n+3)+(-n^2-8*n-16)*a6(n+4), a6(0) = 1, a6(1) = 0, a6(2) = 6, a6(3) = 12}, a6(n),remember): S:= add(U6(n)*x^n/n!,n=0..N)^2: seq(coeff(S,x,n)*n!,n=0..N); # Robert Israel, Nov 15 2016 -
Mathematica
terms = 21; g[x_] = BesselI[0, 2x]^3 + 2 Sum[BesselI[k, 2x]^3, {k, 1, terms}]; CoefficientList[g[x]^2 + O[x]^terms, x] Range[0, terms-1]! (* Jean-François Alcover, Sep 18 2018, after Robert Israel *) -
PARI
seq(n)={Vec(serlaplace(sum(k=0, n, if(k,2,1)*(x^k*besseli(k, 2*x + O(x^(n-k+1)))/k!)^3)^2))} \\ Andrew Howroyd, Nov 01 2018
Formula
E.g.f.: g(x)^2, where g(x) is the e.g.f. of A002898. - Robert Israel, Nov 15 2016
a(n) ~ 2^(2*n) * 3^(n+1) / (Pi^2 * n^2). - Vaclav Kotesovec, Apr 30 2024
Comments