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

1,2

Links

Table of n, a(n) for n=1..40.

Eric Weisstein's World of Mathematics, King Graph

Eric Weisstein's World of Mathematics, Total Dominating Set

Index entries for linear recurrences with constant coefficients, signature (0,0,0,12,0,0,0,-48,0,0,0,64).

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.

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 *)

Prog

(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))}

Crossrefs

Row 2 of A303335.

Cf. A350816.

Sequence in context: A131691 A258504 A273816 * A021063 A216638 A110649

Adjacent sequences: A350814 A350815 A350816 * A350818 A350819 A350820

Keyword

nonn,easy

Author

Andrew Howroyd, Jan 17 2022

Status

approved