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%I #28 Mar 04 2024 00:30:06
%S 0,0,5568,805120,77137920,6307700736,475634876416,34155559059456,
%T 2374835654098944,161464681410592768,10803965184146669568,
%U 714596279034260226048,46865991748990252613632,3054517201890713310068736,198161437014738390623453184
%N Number of 3-cycles in the n-Keller graph.
%H Andrew Howroyd, <a href="/A300818/b300818.txt">Table of n, a(n) for n = 1..100</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GraphCycle.html">Graph Cycle</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/KellerGraph.html">Keller Graph</a>
%H <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (244, -25104, 1434176, -50448128, 1145183232, -17085018112, 167078576128, -1046232170496, 3986606260224, -8340356726784, 7305739370496).
%F a(n) = 2^(2*n - 1)*(3^(2*n + 1) - 3^(n + 1)*4^n - 7^n + 16^n + (4*3^n - 3*4^n + 3*n - 3)*n)/3. - _Eric W. Weisstein_, Mar 20 2018
%F G.f.: 64*x^3*(87 - 8648*x + 319808*x^2 - 4495488*x^3 - 11911936*x^4 + 941117440*x^5 - 8636104704*x^6 + 25777668096*x^7) / ((1 - 4*x)^3*(1 - 12*x)^2*(1 - 16*x)^2*(1 - 28*x)*(1 - 36*x)*(1 - 48*x)*(1 - 64*x)). - _Colin Barker_, Mar 20 2018
%t Table[2^(2 n - 1) (3^(2 n + 1) - 3^(n + 1) 4^n - 7^n + 16^n + (4 3^n - 3 4^n + 3 n - 3) n)/3, {n, 20}]
%t LinearRecurrence[{244, -25104, 1434176, -50448128, 1145183232, -17085018112, 167078576128, -1046232170496, 3986606260224, -8340356726784, 7305739370496}, {0, 0, 5568, 805120, 77137920, 6307700736, 475634876416, 34155559059456, 2374835654098944, 161464681410592768, 10803965184146669568}, 20]
%t CoefficientList[Series[-64 x^2 (87 - 8648 x + 319808 x^2 - 4495488 x^3 - 11911936 x^4 + 941117440 x^5 - 8636104704 x^6 + 25777668096 x^7)/((-1 + 4 x)^3 (-1 + 12 x)^2 (-1 + 16 x)^2 (-1 + 28 x) (-1 + 36 x) (-1 + 48 x) (-1 + 64 x)), {x, 0, 20}], x]
%o (PARI)
%o G(n,k,bset)={
%o my(r=vector(n), m=sum(i=0,k-1,1<<(2*i)), u=vector(1<<(2*k),i,i==1));
%o for(n=1, n,
%o my(v=vector(#u));
%o for(b=1, #bset,
%o my(d=bitxor(bset[b],bset[b]<<2), t1=bitand(d,m), t2=bitand((d>>1),m-t1));
%o for(i=0, #u-1,
%o my(k=bitor(bitor(i, t1), bitor(t2<<1, bitand(i>>1,t2))));
%o v[k+1] += u[i+1] ));
%o u = v;
%o r[n] = 4^n*u[#u]/(2*k) );
%o r
%o }
%o G(10,3,[0..15]) \\ _Andrew Howroyd_, Mar 14 2018
%o (PARI) concat(vector(2), Vec(64*x^3*(87 - 8648*x + 319808*x^2 - 4495488*x^3 - 11911936*x^4 + 941117440*x^5 - 8636104704*x^6 + 25777668096*x^7) / ((1 - 4*x)^3*(1 - 12*x)^2*(1 - 16*x)^2*(1 - 28*x)*(1 - 36*x)*(1 - 48*x)*(1 - 64*x)) + O(x^20))) \\ _Colin Barker_, Mar 20 2018
%o (Magma) [2^(2*n-1)*(3^(2*n+1)-3^(n+1)*4^n-7^n+16^n+(4*3^n- 3*4^n+3*n-3)* n)/3: n in [1..20]]; // _Vincenzo Librandi_, Mar 21 2018
%Y Cf. A300842 (4-cycles), A300848 (5-cycles), A300849 (6-cycles).
%K nonn
%O 1,3
%A _Eric W. Weisstein_, Mar 13 2018
%E Terms a(7) and beyond from _Andrew Howroyd_, Mar 14 2018
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