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A339851 Number of Hamiltonian circuits within parallelograms of size 4 X n on the triangular lattice.

%I A339851 #34 Mar 30 2023 16:40:57
%S A339851 1,13,80,549,3851,26499,183521,1269684,8782833,60764640,420375910,
%T A339851 2908245096,20119820809,139192751951,962962619849,6661962019139,
%U A339851 46088745527485,318850883829314,2205872265781839,15260652269262421,105576152878533354,730396306808551777,5053023343572544589
%N A339851 Number of Hamiltonian circuits within parallelograms of size 4 X n on the triangular lattice.
%H A339851 Seiichi Manyama, <a href="/A339851/b339851.txt">Table of n, a(n) for n = 2..1000</a>
%H A339851 M. Peto, <a href="https://doi.org/10.31274/rtd-180813-17105">Studies of protein designability using reduced models</a>, Thesis, 2007.
%H A339851 <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (3,21,44,-5,-47,-26,83,-81,39,-10,1)
%F A339851 a(n) = 3*a(n-1) + 21*a(n-2) + 44*a(n-3) - 5*a(n-4) - 47*a(n-5) - 26*a(n-6) + 83*a(n-7) - 81*a(n-8) + 39*a(n-9) - 10*a(n-10) + a(n-11) for n > 12.
%F A339851 G.f.: x^2*(1 + 10*x + 20*x^2 - 8*x^3 - 43*x^4 + 9*x^5 + 34*x^6 - 42*x^7 + 24*x^8 - 7*x^9 + x^10) / (1 - 3*x - 21*x^2 - 44*x^3 + 5*x^4 + 47*x^5 + 26*x^6 - 83*x^7 + 81*x^8 - 39*x^9 + 10*x^10 - x^11). - _Vaclav Kotesovec_, Dec 23 2020
%t A339851 CoefficientList[Series[x^2(1+10x+20x^2-8x^3-43x^4+9x^5+34x^6-42x^7+24x^8-7x^9+x^10)/(1-3x-21x^2-44x^3+5x^4+47x^5+26x^6-83x^7+81x^8-39x^9+10x^10-x^11),{x,0,30}],x] (* or *) LinearRecurrence[{3,21,44,-5,-47,-26,83,-81,39,-10,1},{1,13,80,549,3851,26499,183521,1269684,8782833,60764640,420375910},30] (* _Harvey P. Dale_, Mar 30 2023 *)
%o A339851 (PARI) N=40; a=vector(N); a[2]=1; a[3]=13; a[4]=80; a[5]=549; a[6]=3851; a[7]=26499; a[8]=183521; a[9]=1269684; a[10]=8782833; a[11]=60764640; a[12]=420375910; for(n=13, N, a[n]=3*a[n-1]+21*a[n-2]+44*a[n-3]-5*a[n-4]-47*a[n-5]-26*a[n-6]+83*a[n-7]-81*a[n-8]+39*a[n-9]-10*a[n-10]+a[n-11]); a[2..N]
%o A339851 (Python)
%o A339851 # Using graphillion
%o A339851 from graphillion import GraphSet
%o A339851 def make_T_nk(n, k):
%o A339851     grids = []
%o A339851     for i in range(1, k + 1):
%o A339851         for j in range(1, n):
%o A339851             grids.append((i + (j - 1) * k, i + j * k))
%o A339851             if i < k:
%o A339851                 grids.append((i + (j - 1) * k, i + j * k + 1))
%o A339851     for i in range(1, k * n, k):
%o A339851         for j in range(1, k):
%o A339851             grids.append((i + j - 1, i + j))
%o A339851     return grids
%o A339851 def A339849(n, k):
%o A339851     universe = make_T_nk(n, k)
%o A339851     GraphSet.set_universe(universe)
%o A339851     cycles = GraphSet.cycles(is_hamilton=True)
%o A339851     return cycles.len()
%o A339851 def A339851(n):
%o A339851     return A339849(4, n)
%o A339851 print([A339851(n) for n in range(2, 21)])
%Y A339851 Row 4 of A339849.
%Y A339851 Cf. A339201.
%K A339851 nonn
%O A339851 2,2
%A A339851 _Seiichi Manyama_, Dec 19 2020