OFFSET
1,2
COMMENTS
Sequence extrapolated to n=1 using formula. - Andrew Howroyd, May 20 2018
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Antiprism Graph
Eric Weisstein's World of Mathematics, Maximum Irredundant Vertex Set
Index entries for linear recurrences with constant coefficients, signature (0,0,5,0,0,-10,0,0,10,0,0,-5,0,0,1).
FORMULA
From Andrew Howroyd, May 21 2018: (Start)
a(n) = 5*a(n-3) - 10*a(n-6) + 10*a(n-9) - 5*a(n-12) + a(n-15) for n > 15.
a(3*k) = 15, a(3*k+1) = (3*k+1)*(k+1)*(5*k^2+10*k+6)/6, a(3*k+2) = 2*(k+1)*(3*k+2). (End)
G.f.: x*(1 + 4*x + 15*x^2 + 23*x^3 - 60*x^5 + 31*x^6 - 12*x^7 + 90*x^8 + 5*x^9 + 8*x^10 - 60*x^11 + 15*x^14) / ((1 - x)^5*(1 + x + x^2)^5). - Colin Barker, May 22 2018
MATHEMATICA
Table[Piecewise[{{15, Mod[n, 3] == 0}, {n (n + 2) (29 + 20 n + 5 n^2)/162, Mod[n, 3] == 1}, {2 n (n + 1)/3, Mod[n, 3] == 2}}], {n, 20}]
LinearRecurrence[{0, 0, 5, 0, 0, -10, 0, 0, 10, 0, 0, -5, 0, 0, 1}, {1, 4, 15, 28, 20, 15, 161, 48, 15, 540, 88, 15, 1365, 140, 15}, 20]
Table[(2430 + 166 n + 177 n^2 + 30 n^3 + 5 n^4 - (-4860 + 166 n + 177 n^2 + 30 n^3 + 5 n^4) Cos[2 n Pi/3] + Sqrt[3] n (-50 - 39 n + 30 n^2 + 5 n^3) Sin[2 n Pi/3])/486, {n, 20}]
CoefficientList[Series[15/(1 - x^3) x^2 - (1 + 23 x^3 + 31 x^6 + 5 x^9)/(-1 + x^3)^5 - (4 x (1 + 2 x^3))/(-1 + x^3)^3, {x, 0, 20}], x]
PROG
(PARI) a(n)={if(n%3==0, 15, my(k=n\3); n*(k+1)*if(n%3==1, (5*k^2+10*k+6)/6, 2))} \\ Andrew Howroyd, May 20 2018
(PARI) Vec(x*(1 + 4*x + 15*x^2 + 23*x^3 - 60*x^5 + 31*x^6 - 12*x^7 + 90*x^8 + 5*x^9 + 8*x^10 - 60*x^11 + 15*x^14) / ((1 - x)^5*(1 + x + x^2)^5) + O(x^50)) \\ Colin Barker, May 22 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Eric W. Weisstein, May 14 2018
EXTENSIONS
a(1)-a(2) and terms a(14) and beyond from Andrew Howroyd, May 20 2018
STATUS
approved