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

1,2

Links

Table of n, a(n) for n=1..52.

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).

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)

Example

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
     . . . .
.

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 *)

Prog

(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

Crossrefs

Cf. A003215, A049450.

Partial sums of A133090.

Sequence in context: A240469 A257335 A152211 * A125852 A368824 A336903

Adjacent sequences: A309802 A309803 A309804 * A309806 A309807 A309808

Keyword

nonn,easy

Author

Sangeet Paul, Aug 17 2019

Status

approved