OFFSET
0,3
COMMENTS
The sequence has been extended to n=0 using the recurrence. - Andrew Howroyd, Dec 18 2024
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..200
Eric Weisstein's World of Mathematics, Double Cone Graph.
Eric Weisstein's World of Mathematics, Minimal Edge Cover.
Index entries for linear recurrences with constant coefficients, signature (4,0,-2,-21,8,8,34,-12,20,-28,6,-24,20,-13,12,-4,2,-1).
FORMULA
G.f.: x*(1 + 17*x - 26*x^2 - 81*x^3 + 33*x^4 - 39*x^5 + 118*x^6 - 312*x^7 + 461*x^8 - 260*x^9 + 183*x^10 - 211*x^11 + 33*x^12 + 10*x^13 + 3*x^14 - 2*x^16)/((1 + x + x^2 - x^3)*(1 + x - x^3)^2*(1 - 2*x + x^2 - x^3)^2*(1 - 3*x - x^2 - x^3)). - Andrew Howroyd, Dec 18 2024
a(n) = 4*a(n-1) - 2*a(n-3) - 21*a(n-4) + 8*a(n-5) + 8*a(n-6) + 34*a(n-7) - 12*a(n-8) + 20*a(n-9) - 28*a(n-10) + 6*a(n-11) - 24*a(n-12) + 20*a(n-13) - 13*a(n-14) + 12*a(n-15) - 4*a(n-16) + 2*a(n-17) - a(n-18). - Eric W. Weisstein, Oct 01 2025
MATHEMATICA
LinearRecurrence[{4, 0, -2, -21, 8, 8, 34, -12, 20, -28, 6, -24, 20, -13, 12, -4, 2, -1}, {0, 1, 21, 58, 149, 566, 1676, 5482, 18021, 59665, 199700, 670517, 2259384, 7624878, 25759564, 87078065, 294452965, 995889190}, 20] (* Eric W. Weisstein, Oct 01 2025 *)
CoefficientList[Series[x (1 + 17 x - 26 x^2 - 81 x^3 + 33 x^4 - 39 x^5 + 118 x^6 - 312 x^7 + 461 x^8 - 260 x^9 + 183 x^10 - 211 x^11 + 33 x^12 + 10 x^13 + 3 x^14 - 2 x^16)/((1 - 2 x - 3 x^2 - 6 x^3 + x^4 + x^6) (1 - x - x^2 - x^3 + x^4 - x^5 + x^6)^2), {x, 0, 20}], x] (* Eric W. Weisstein, Oct 01 2025 *)
PROG
(PARI) seq(n)={my(g1 = 1/(1-x -x^2 - x^3) + O(x*x^n), g2 = 1/(1-x^2-x^3) + O(x*x^n), h1 = g1 + x^2*g1 + 2*x^3*g1, h2 = g2 + x^2*g2 + 2*x^3*g2); Vec(serconvol(h1, h1) - serconvol(h2, h2) + 2*serconvol(h2, x*deriv(2*x^2*g2 + x^3*g2)), -n-1)} \\ Andrew Howroyd, Dec 18 2024
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Eric W. Weisstein, Dec 17 2024
EXTENSIONS
a(0)-a(2) prepended and a(8) onwards from Andrew Howroyd, Dec 18 2024
STATUS
approved