cp's OEIS Frontend

A107453 1 followed by repetitions of the period-4 sequence 1,1,1,2.

%I A107453 #45 Dec 14 2023 05:26:14
%S A107453 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,
%T A107453 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,
%U A107453 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
%N A107453 1 followed by repetitions of the period-4 sequence 1,1,1,2.
%C A107453 Number of nonisomorphic generalized Petersen graphs P(n,k) with girth 4 on 2n vertices for 1<=k<=floor((n-1)/2).
%C A107453 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.
%C A107453 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.
%C A107453 From Tomaz Pisanski, Mar 08 2008: (Start)
%C A107453 The fact that the two interpretations give the same numerical values is a coincidence.
%C A107453 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.
%C A107453 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).
%C A107453 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)
%C A107453 Also, Table[Denominator[(n - 1) n (n + 1)/12], {n, 100}] with 3 1's in front... - _Eric W. Weisstein_, Mar 04 2008
%C A107453 Continued fraction expansion of sqrt(8/3), if the offset is 1. - _Arkadiusz Wesolowski_, Aug 27 2011
%D A107453 I. Z. Bouwer, W. W. Chernoff, B. Monson and Z. Star, The Foster Census (Charles Babbage Research Centre, 1988), ISBN 0-919611-19-2.
%H A107453 Marko Boben, Tomaz Pisanski and Arjana Zitnik, <a href="http://preprinti.imfm.si/PDF/00939.pdf">I-graphs and the corresponding configurations</a>, Preprint series (University of Ljubljana, IMFM), Vol. 42 (2004), 939 (ISSN 1318-4865).
%H A107453 M. Watkins, <a href="https://doi.org/10.1016/S0021-9800(69)80116-X">A theorem on Tait colorings with an application to the generalized Petersen graphs</a>, J. Combin. Theory 6 (1969), 152-164.
%H A107453 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,1).
%F A107453 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
%F A107453 From _Colin Barker_, Jul 16 2013: (Start)
%F A107453 a(n) = (5+(-1)^n+(-i)^n+i^n)/4 for n>4, where i=sqrt(-1).
%F A107453 G.f.: -x^4*(x^4+x^3+x^2+x+1) / ((x-1)*(x+1)*(x^2+1)). (End)
%e A107453 A generalized Petersen graph P(n,k) has girth 4 if and only if n = 4k or k=1.
%e A107453 The smallest generalized Petersen graph with girth 4 is P(4,1).
%e A107453 The smallest bipartite generalized Petersen graph with girth 4 is P(4,1).
%t A107453 Join[{1},PadRight[{},104,{1,1,1,2}]] (* _Harvey P. Dale_, Oct 25 2011 *)
%o A107453 (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
%Y A107453 Cf. A077105, A107452, A107454-A107457, A107459, A107460.
%K A107453 nonn,easy
%O A107453 4,5
%A A107453 Marko Boben (Marko.Boben(AT)fmf.uni-lj.si), _Tomaz Pisanski_ and Arjana Zitnik (Arjana.Zitnik(AT)fmf.uni-lj.si), May 26 2005
%E A107453 Edited by _N. J. A. Sloane_, Mar 08 2008