A072266 -id:A072266 - cp's OEIS frontend

A377573 Cogrowth sequence for the 14-element dihedral group D7 = .

1, 0, 1, 0, 3, 0, 10, 1, 35, 9, 126, 55, 462, 286, 1717, 1365, 6451, 6188, 24463, 27132, 93518, 116281, 360031, 490337, 1394582, 2043275, 5430530, 8439210, 21242341, 34621041, 83411715, 141290436, 328589491, 574274008, 1297937234, 2326683921, 5138431851

Offset: 0

Sean A. Irvine, Nov 01 2024

Taking the overlay of the two generating functions in the bisections A072844 and A072266, shows that a(n) = A094052(n-1), n>0. - R. J. Mathar, Nov 05 2024

			a(4) = 3 corresponds to the TTTT = TSTS = STST = 1. Note: TSTS = (TSTS)(TT) = T(STST)T = TT = 1.
a(9) = 9 corresponds to the words SSSSSSSTT = SSSSSSTTS = SSSSSTTSS = SSSSTTSSS = SSSTTSSSS = SSTTSSSSS = STTSSSSSS = TTSSSSSSS = TSSSSSSST = 1.

Sean A. Irvine, 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.
Haggai Liu, Enumerative Properties of Cogrowth Series on Free Products of Finite Groups, ACA 2021 Session on Algorithmic Combinatorics, 2021.
Sean A. Irvine, Java program (github)

Bisections: A072266, A072844.

Cf. A052964 (D5), A007583 (D6), A007582 (D8).

G.f.: F_7(x) where F_n(x) = 1/2 + (1/(2*n)) * Sum_{j=0..n-1} 1 / (1 - 2*cos(2*Pi*j/n)*x).

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

Jamaine Paddyfoot and John W. Layman, Jul 24 2002

nonn

			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).

Haggai Liu, Enumerative Properties of Cogrowth Series on Free Products of Finite Groups, ACA 2021 Session on Algorithmic Combinatorics, 2021.

Cf. A072266.

Bisection of A377573.

a(n) = 9*a(n-1) - 26*a(n-2) + 25*a(n-3) - 4*a(n-4).
g.f.: x^4 / ((1 - 4*x)*(1 - 5*x + 6*x^2 - x^3)). - Colin Barker, Feb 24 2017
28*a(n) = 4^n -4*( 2*A005021(n) -9*A005021(n-1) +11*A005021(n-2) ). - R. J. Mathar, Nov 05 2024

cp's OEIS Frontend

A377573 Cogrowth sequence for the 14-element dihedral group D7 = .

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