OFFSET
1,1
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, Minimum Dominating Set
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,5,0,0,0,0,-10,0,0,0,0,10,0,0,0,0,-5,0,0,0,0,1).
FORMULA
From Andrew Howroyd, May 20 2018: (Start)
a(n) = 5*a(n-5) - 10*a(n-10) + 10*a(n-15) - 5*a(n-20) + a(n-25) for n > 25.
a(5*k) = 5, a(5*k+1) = 2*(5*k+1)*(k+1)*(2*k+3)/3, a(5*k+2) = 2*(5*k+2), a(5*k+3) = (5*k+3)*(2*k+5)*(2*k+4)*(2*k+3)/12, a(5*k+4)=(5*k+4)*(2*k+3). (End)
G.f.: x*(2 + 4*x + 15*x^2 + 12*x^3 + 5*x^4 + 30*x^5 - 6*x^6 + 65*x^7 - 15*x^8 - 20*x^9 - 26*x^10 - 6*x^11 - 4*x^12 - 7*x^13 + 30*x^14 - 6*x^15 + 14*x^16 + 5*x^17 + 11*x^18 - 20*x^19 - 6*x^21 - x^22 - x^23 + 5*x^24) / ((1 - x)^5*(1 + x + x^2 + x^3 + x^4)^5). - Colin Barker, May 22 2018
MATHEMATICA
Table[Piecewise[{{5, Mod[n, 5] == 0}, {2 n (4 + n) (13 + 2 n)/75, Mod[n, 5] == 1}, {2 n, Mod[n, 5] == 2}, {n (7 + n) (9 + 2 n) (19 + 2 n)/750, Mod[n, 5] == 3}, {n (7 + 2 n)/5, Mod[n, 5] == 4}}], {n, 30}]
LinearRecurrence[{0, 0, 0, 0, 5, 0, 0, 0, 0, -10, 0, 0, 0, 0, 10, 0, 0, 0, 0, -5, 0, 0, 0, 0, 1}, {2, 4, 15, 12, 5, 40, 14, 140, 45, 5, 154, 24, 546, 98, 5, 384, 34, 1485, 171, 5, 770, 44, 3289, 264, 5}, 30]
CoefficientList[Series[(5 x^4)/(1 - x^5) + (2 x (2 + 3 x^5))/(-1 + x^5)^2 + (x^3 (-12 - 9 x^5 + x^10))/(-1 + x^5)^3 + (2 (1 + 16 x^5 + 3 x^10))/(-1 + x^5)^4 + (x^2 (-15 - 65 x^5 + 4 x^10 - 5 x^15 + x^20))/(-1 + x^5)^5, {x, 0, 30}], x]
PROG
(PARI) a(n)={[k->5, k->2*(5*k+1)*(k+1)*(2*k+3)/3, k->2*(5*k+2), k->(5*k+3)*(2*k+5)*(2*k+4)*(2*k+3)/12, k->(5*k+4)*(2*k+3)][n%5+1](n\5)} \\ Andrew Howroyd, May 20 2018
(PARI) Vec(x*(2 + 4*x + 15*x^2 + 12*x^3 + 5*x^4 + 30*x^5 - 6*x^6 + 65*x^7 - 15*x^8 - 20*x^9 - 26*x^10 - 6*x^11 - 4*x^12 - 7*x^13 + 30*x^14 - 6*x^15 + 14*x^16 + 5*x^17 + 11*x^18 - 20*x^19 - 6*x^21 - x^22 - x^23 + 5*x^24) / ((1 - x)^5*(1 + x + x^2 + x^3 + x^4)^5) + O(x^40)) \\ Colin Barker, May 22 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Eric W. Weisstein, May 14 2018
EXTENSIONS
a(1)-a(2) and terms a(21) and beyond from Andrew Howroyd, May 20 2018
STATUS
approved