A087327 Independence numbers for KT_2 knight on triangular board.

1, 2, 6, 8, 13, 18, 25, 32, 41, 50, 61, 72, 85, 98, 113, 128, 145, 162, 181, 200, 221, 242, 265, 288, 313, 338, 365, 392, 421, 450, 481, 512, 545, 578, 613, 648, 685, 722, 761, 800, 841, 882, 925, 968, 1013, 1058, 1105, 1152, 1201, 1250, 1301, 1352, 1405, 1458

Offset

1,2

Links

Colin Barker, Table of n, a(n) for n = 1..1000

J.-P. Bode and H. Harborth, Independence for knights on hexagon and triangle boards, Discrete Math., 272 (2003), 27-35.

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

Formula

a(n) = ceiling(n^2/2) except for n=3.

From Colin Barker, Feb 02 2016: (Start)

a(n) = (2*n^2-(-1)^n+1)/4 for n>3.

a(n) = n^2/2 for even n>3; a(n) = (n^2+1)/2 for odd n>3.

a(n) = 2*a(n-1)-2*a(n-3)+a(n-4) for n>3.

G.f.: x*(1+2*x^2-2*x^3+2*x^5-x^6) / ((1-x)^3*(1+x)). (End)

Mathematica

LinearRecurrence[{2, 0, -2, 1}, {1, 2, 6, 8, 13, 18, 25}, 60] (* Harvey P. Dale, Mar 14 2018 *)

Prog

(PARI) Vec(x*(1+2*x^2-2*x^3+2*x^5-x^6)/((1-x)^3*(1+x)) + O(x^100)) \\ Colin Barker, Feb 02 2016

Crossrefs

Cf. A000982, A030978, A087328, A087329.

Sequence in context: A038108 A356217 A294862 * A266627 A289753 A002511

Adjacent sequences: A087324 A087325 A087326 * A087328 A087329 A087330

Keyword

nonn,easy

Author

N. J. A. Sloane, Oct 21 2003

Extensions

More terms from David Wasserman, May 06 2005

Status

approved