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A107453
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1 followed by repetitions of the period-4 sequence 1,1,1,2.
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4
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1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2
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OFFSET
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4,5
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COMMENTS
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Number of nonisomorphic generalized Petersen graphs P(n,k) with girth 4 on 2n vertices for 1<=k<=floor((n-1)/2).
The generalized Petersen graph P(n,k) is a graph with vertex set V(P(n,k)) = {u_0,u_1,...,u_{n-1},v_0,v_1,...,v_{n-1}} and edge set E(P(n,k)) = {u_i u_{i+1}, u_i v_i, v_i v_{i+k} : i=0,...,n-1}, where the subscripts are to be read modulo n.
Also the number of nonisomorphic bipartite generalized Petersen graphs P(2n,k) with girth 4 on 4n vertices for 1<=k<n, n >= 2. A generalized Petersen graph P(n,k) is bipartite if and only if n is even and k is odd; it has girth 4 if and only if n = 4k or k=1.
From Tomaz Pisanski, Mar 08 2008: (Start)
The fact that the two interpretations give the same numerical values is a coincidence.
Let f(n) be the number of generalized Petersen graphs P(n,k), n = 4,5,... of girth 4. Let g(n) be the number of bipartite generalized Petersen graphs P(2n,k), n = 2,3,4,... of girth 4.
The sequences may be computed as follows: f(t) = if t = 4 then 1 else if 4|t then 2 else 1 and g(s) = if s = 2 then else if mod(s,4) = 2 then 2 else 1. It follows that f(n+2) = g(n).
The exception f(4) = g(2) = 1 does count the same object, namely, P(4,1) but for all other cases f(n+2) counts different objects that g(n). (End)
Also, Table[Denominator[(n - 1) n (n + 1)/12], {n, 100}] with 3 1's in front... - Eric W. Weisstein, Mar 04 2008
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REFERENCES
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I. Z. Bouwer, W. W. Chernoff, B. Monson and Z. Star, The Foster Census (Charles Babbage Research Centre, 1988), ISBN 0-919611-19-2.
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LINKS
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FORMULA
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a(n) = sgn(n) + cos(Pi*n/4)^2 + (cos(Pi*n)-1)/4; a(n) = sgn(n) + floor(((n+3) mod 4)/3). - Carl R. White, Oct 15 2009
a(n) = (5+(-1)^n+(-i)^n+i^n)/4 for n>4, where i=sqrt(-1).
G.f.: -x^4*(x^4+x^3+x^2+x+1) / ((x-1)*(x+1)*(x^2+1)). (End)
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EXAMPLE
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A generalized Petersen graph P(n,k) has girth 4 if and only if n = 4k or k=1.
The smallest generalized Petersen graph with girth 4 is P(4,1).
The smallest bipartite generalized Petersen graph with girth 4 is P(4,1).
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MATHEMATICA
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Join[{1}, PadRight[{}, 104, {1, 1, 1, 2}]] (* Harvey P. Dale, Oct 25 2011 *)
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PROG
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(PARI) x='x+O('x^100); Vec(-x^4*(x^4+x^3+x^2+x+1)/((x-1)*(x+1)*(x^2+1))) \\ Altug Alkan, Dec 24 2015
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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Marko Boben (Marko.Boben(AT)fmf.uni-lj.si), Tomaz Pisanski and Arjana Zitnik (Arjana.Zitnik(AT)fmf.uni-lj.si), May 26 2005
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EXTENSIONS
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STATUS
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approved
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