OFFSET
1,2
LINKS
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
KEYWORD
nonn,easy
AUTHOR
Andrew Howroyd, Jan 17 2022
STATUS
approved