A095364
Number of walks of length n between two adjacent nodes in the cycle graph C_9.
Original entry on oeis.org
1, 0, 3, 0, 10, 0, 35, 1, 126, 11, 462, 78, 1716, 455, 6435, 2380, 24311, 11628, 92398, 54264, 352947, 245157, 1354102, 1081575, 5215250, 4686826, 20156580, 20030039, 78152535, 84672780, 303906051, 354822776, 1184959314, 1476390160
Offset: 1
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a(n) = round(2^n/9*sum(r=0, 8, cos(2*Pi*r/9)^(n+1))) \\ Michel Marcus, Jul 18 2013
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Vec( x*(-1+x+2*x^2-x^3)/((1+x)*(-1+2*x)*(1-3*x^2+x^3))+O(x^66) ) \\ Joerg Arndt, Jul 18 2013
A072266
Number of words of length 2n generated by the two letters s and t that reduce to the identity 1 using the relations sssssss=1, tt=1 and stst=1. The generators s and t along with the three relations generate the 14-element dihedral group D7.
Original entry on oeis.org
1, 1, 3, 10, 35, 126, 462, 1717, 6451, 24463, 93518, 360031, 1394582, 5430530, 21242341, 83411715, 328589491, 1297937234, 5138431851, 20380608990, 80960325670, 322016144629, 1282138331587, 5109310929719, 20374764059254
Offset: 0
Jamaine Paddyfoot (jay_paddyfoot(AT)hotmail.com) and John W. Layman, Jul 08 2002
The words tttt=tsts=stst=1 so a(2)=3.
- Colin Barker, Table of n, a(n) for n = 0..1000
- H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, 4th ed., Springer-Verlag, NY, reprinted 1984, p. 134.
- Index entries for linear recurrences with constant coefficients, signature (9,-26,25,-4).
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LinearRecurrence[{9,-26,25,-4},{1,1,3,10,35},30] (* Harvey P. Dale, Apr 16 2022 *)
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a(n)=if(n<1,n==0,sum(k=-(n-1)\7,(n-1)\7,C(2*n-1,n+7*k)))
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Vec((1 - 8*x + 20*x^2 - 16*x^3 + 2*x^4) / ((1 - 4*x)*(1 - 5*x + 6*x^2 - x^3)) + O(x^30)) \\ Colin Barker, Apr 26 2019
A072844
Number of words of length 2n-1 generated by the two letters s and t that reduce to the identity 1 by using the relations sssssss=1, tt=1 and stst=1. The generators s and t along with the three relations generate the 14-element dihedral group D7.
Original entry on oeis.org
0, 0, 0, 1, 9, 55, 286, 1365, 6188, 27132, 116281, 490337, 2043275, 8439210, 34621041, 141290436, 574274008, 2326683921, 9402807817, 37923176863, 152705590518, 614111175965, 2467123420524, 9903167265124, 39725253489545
Offset: 1
The 9 words of length 9 are ssssssstt, sssssstts, sssssttss, ssssttsss, sssttssss, ssttsssss, sttssssss, ttsssssss, tssssssst. - _Sean A. Irvine_, Oct 31 2024
- H.S.M. Coxeter and W.O.J. Moser, Generators and Relations for Discrete Groups, Fourth Edition, (p.134).
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