A350817 Number of minimum total dominating sets in the 2 X n king graph.
1, 6, 9, 4, 8, 89, 56, 16, 64, 780, 304, 64, 384, 5472, 1536, 256, 2048, 33920, 7424, 1024, 10240, 194304, 34816, 4096, 49152, 1053696, 159744, 16384, 229376, 5488640, 720896, 65536, 1048576, 27721728, 3211264, 262144, 4718592, 136642560, 14155776, 1048576
Offset: 1
Links
- Eric Weisstein's World of Mathematics, King Graph.
- Eric Weisstein's World of Mathematics, Minimum Total Dominating Set.
- Index entries for linear recurrences with constant coefficients, signature (0,0,0,12,0,0,0,-48,0,0,0,64).
Programs
-
Mathematica
LinearRecurrence[{0, 0, 0, 12, 0, 0, 0, -48, 0, 0, 0, 64}, {1, 6, 9, 4, 8, 89, 56, 16, 64, 780, 304, 64, 384}, 40] (* Michael De Vlieger, Jan 19 2022 *) -
PARI
Vec((1 + 6*x + 9*x^2 + 4*x^3 - 4*x^4 + 17*x^5 - 52*x^6 - 32*x^7 + 16*x^8 + 64*x^10 + 64*x^11 - 64*x^12)/((1 - 2*x^2)^3*(1 + 2*x^2)^3) + O(x^40))
-
PARI
a(n)={my(k=n\4); 4^k*if(n%2, if(n%4==1, (k==0) + 2*k, 5*k + 9), if(n%4==0, 1, (k + 1)*(41*k + 48)/8))}
Formula
a(n) = 12*a(n-4) - 48*a(n-8) + 64*a(n-12) for n > 13.
G.f.: x*(1 + 6*x + 9*x^2 + 4*x^3 - 4*x^4 + 17*x^5 - 52*x^6 - 32*x^7 + 16*x^8 + 64*x^10 + 64*x^11 - 64*x^12)/((1 - 2*x^2)^3*(1 + 2*x^2)^3).
a(4*k) = 4^k; a(4*k+1) = 2*k*4^k for k > 0; a(4*k+2) = (k + 1)*(41*k + 48)*4^k/8; a(4*k+3) = (5*k + 9)*4^k.