A300818 Number of 3-cycles in the n-Keller graph.

0, 0, 5568, 805120, 77137920, 6307700736, 475634876416, 34155559059456, 2374835654098944, 161464681410592768, 10803965184146669568, 714596279034260226048, 46865991748990252613632, 3054517201890713310068736, 198161437014738390623453184

Offset

1,3

Links

Andrew Howroyd, Table of n, a(n) for n = 1..100

Eric Weisstein's World of Mathematics, Graph Cycle

Eric Weisstein's World of Mathematics, Keller Graph

Index entries for linear recurrences with constant coefficients, signature (244, -25104, 1434176, -50448128, 1145183232, -17085018112, 167078576128, -1046232170496, 3986606260224, -8340356726784, 7305739370496).

Formula

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

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

Mathematica

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}]
LinearRecurrence[{244, -25104, 1434176, -50448128, 1145183232, -17085018112, 167078576128, -1046232170496, 3986606260224, -8340356726784, 7305739370496}, {0, 0, 5568, 805120, 77137920, 6307700736, 475634876416, 34155559059456, 2374835654098944, 161464681410592768, 10803965184146669568}, 20]
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]

Prog

(PARI)
G(n, k, bset)={
my(r=vector(n), m=sum(i=0, k-1, 1<<(2*i)), u=vector(1<<(2*k), i, i==1));
for(n=1, n,
  my(v=vector(#u));
  for(b=1, #bset,
    my(d=bitxor(bset[b], bset[b]<<2), t1=bitand(d, m), t2=bitand((d>>1), m-t1));
    for(i=0, #u-1,
      my(k=bitor(bitor(i, t1), bitor(t2<<1, bitand(i>>1, t2))));
      v[k+1] += u[i+1] ));
  u = v;
  r[n] = 4^n*u[#u]/(2*k) );
r
}
G(10, 3, [0..15]) \\ Andrew Howroyd, Mar 14 2018
(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
(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

Crossrefs

Cf. A300842 (4-cycles), A300848 (5-cycles), A300849 (6-cycles).

Sequence in context: A043512 A167281 A252907 * A252154 A360218 A065698

Adjacent sequences: A300815 A300816 A300817 * A300819 A300820 A300821

Keyword

nonn

Author

Eric W. Weisstein, Mar 13 2018

Extensions

Terms a(7) and beyond from Andrew Howroyd, Mar 14 2018

Status

approved