A082873 Independence number of king KG_2 on triangle board B_n.

1, 1, 3, 6, 8, 12, 15, 19, 25, 30, 36, 42, 49, 55, 63, 72, 80, 90, 99, 109, 121, 132, 144, 156, 169, 181, 195, 210, 224, 240, 255, 271, 289, 306, 324, 342, 361, 379, 399, 420, 440, 462, 483, 505, 529, 552, 576, 600, 625, 649, 675, 702, 728

Offset

1,3

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

J.-J. Bode, H. Harborth and M. Harborth, King independence on triangle boards, Discr. Math., 266 (2003), 101-107.

Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,0,0,0,0,0,1,-2,1).

Formula

a(n) = floor((n+1)^2/4) if n == 0, 1, 4, 6, 9, 10, 11 (mod 12), a(n) = floor((n+1)^2/4) - 1 otherwise.

a(n) = +2*a(n-1) -a(n-2) +a(n-12) -2*a(n-13) +a(n-14). - R. J. Mathar, Aug 05 2014

G.f.: -x*(1-x+2*x^2-x^4+2*x^5-x^6+x^7+2*x^8-x^9+x^10+x^3) / ( (1+x) *(1+x^2) *(x^4-x^2+1) *(x^2-x+1) *(1+x+x^2) *(x-1)^3 ). - R. J. Mathar, Aug 05 2014

Mathematica

CoefficientList[Series[-(1 - x + 2 x^2 - x^4 + 2 x^5 - x^6 + x^7 + 2 x^8 - x^9 + x^10 + x^3)/((1 + x) (1 + x^2) (x^4 - x^2 + 1) (x^2 - x + 1) (1 + x + x^2) (x - 1)^3), {x, 0, 40}], x] (* Vincenzo Librandi, Aug 06 2014 *)

Prog

(Magma) I:=[1, 1, 3, 6, 8, 12, 15, 19, 25, 30, 36, 42, 49, 55]; [n le 14 select I[n] else 2*Self(n-1)-Self(n-2)+Self(n-12)-2*Self(n-13)+Self(n-14): n in [1..60]]; // Vincenzo Librandi, Aug 06 2014

Crossrefs

Cf. A082874.

Sequence in context: A338722 A185549 A185597 * A182727 A220001 A310138

Adjacent sequences: A082870 A082871 A082872 * A082874 A082875 A082876

Keyword

nonn,easy,changed

Author

N. J. A. Sloane, May 25 2003

Extensions

More terms from Sean A. Irvine, Sep 26 2011

Status

approved