A309805 Maximum number of nonattacking kings placeable on a hexagonal board with edge-length n in Glinski's hexagonal chess.

1, 2, 7, 10, 19, 24, 37, 44, 61, 70, 91, 102, 127, 140, 169, 184, 217, 234, 271, 290, 331, 352, 397, 420, 469, 494, 547, 574, 631, 660, 721, 752, 817, 850, 919, 954, 1027, 1064, 1141, 1180, 1261, 1302, 1387, 1430, 1519, 1564, 1657, 1704, 1801, 1850, 1951, 2002

Offset: 1

Author

Sangeet Paul, Aug 17 2019

Keywords

• nonn
• easy

Examples

			a(1) = 1
.
  o
.
a(2) = 2
.
   . .
  o . o
   . .
.
a(3) = 7
.
    o . o
   . . . .
  o . o . o
   . . . .
    o . o
.
a(4) = 10
.
     . . . .
    o . o . o
   . . . . . .
  o . o . o . o
   . . . . . .
    o . o . o
     . . . .
.
		

Links

• Chess variants, Glinski's Hexagonal Chess
• Wikipedia, Hexagonal chess - Gliński's hexagonal chess
• Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Crossrefs

Cf. A003215, A049450.

Partial sums of A133090.

Programs

Mathematica

nn:=51; CoefficientList[Series[- (1 + x + 3*x^2 + x^3)/((- 1 + x)^3*(1 + x)^2),{x, 0, nn}], x] (* Georg Fischer, May 10 2020 *)

PARI

a(n) = n^2 - (n\2) - (n\2)^2; \\ Andrew Howroyd, Aug 17 2019

Python

def A309805(n): return n**2-(m:=n>>1)*(m+1) # Chai Wah Wu, Apr 04 2024

Formula

a(n) = n^2 - floor(n/2) - floor(n/2)^2.

From Stefano Spezia, Aug 18 2019 (Start)

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

E.g.f.: (1/8)*exp(-x)*(-1 + 2*x + exp(2*x)*(1 + 4*x + 6*x^2)).

a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n > 5.

a(n) = (1/16)*(3 + (-1)^(1+2*n) - 4*n + 12*n^2 - 2*(-1)^n*(1 + 2*n)).

a(2*n-1) = A003215(n).

a(2*n) = A049450(n).

(End)