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
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).
Programs
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Mathematica
LinearRecurrence[{2,0,-2,1},{1,2,6,8,13,18,25},60] (* Harvey P. Dale, Mar 14 2018 *) -
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
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)
Extensions
More terms from David Wasserman, May 06 2005