A274616 Maximal number of non-attacking queens on a right triangular board with n cells on each side.

0, 1, 1, 2, 2, 3, 4, 5, 5, 6, 7, 7, 8, 9, 9, 10, 11, 11, 12, 13, 13, 14, 15, 15, 16, 17, 17, 18, 19, 19, 20, 21, 21, 22, 23, 23, 24, 25, 25, 26, 27, 27, 28, 29, 29, 30, 31, 31, 32, 33, 33, 34, 35, 35, 36, 37, 37, 38, 39, 39, 40, 41, 41, 42, 43, 43, 44, 45, 45, 46, 47, 47

Offset

0,4

Comments

This sequence was mentioned by R. K. Guy in the first comment in A004396.

References

Paul Vanderlind, Richard K. Guy, and Loren C. Larson, The Inquisitive Problem Solver, MAA, 2002. See Problem 252, pages 67, 87, 198 and 276.

Links

Colin Barker, Table of n, a(n) for n = 0..1000

F. Michel Dekking, Jeffrey Shallit, and N. J. A. Sloane, Queens in exile: non-attacking queens on infinite chess boards, Electronic J. Combin., 27:1 (2020), #P1.52.

Gabriel Nivasch and Eyal Lev, Nonattacking Queens on a Triangle, Mathematics Magazine, Vol. 78, No. 5 (Dec., 2005), pp. 399-403.

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

Formula

Except for n=4, this is round(2n/3).

From Colin Barker, Jul 02 2016: (Start)

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

G.f.: x*(1+x^2-x^3)*(1+x^4)/((1-x)^2*(1+x+x^2)). (End)

a(n) = 2*(3*n + sqrt(3)*sin((2*Pi*n)/3)) / 9. - Colin Barker, Mar 08 2017

Example

n=3:
OOX
XO
O
n=4:
OOOX
OXO
OO
O
n=5:
OOOOX
OOXO
XOO
OO
O

Mathematica

CoefficientList[Series[x*(1 +x^2 -x^3)*(1 +x^4)/((1-x)^2*(1+x+x^2)), {x, 0, 50}], x] (* G. C. Greubel, Jul 03 2016 *)

Prog

(PARI) concat(0, Vec(x*(1+x^2-x^3)*(1+x^4)/((1-x)^2*(1+x+x^2)) + O(x^100))) \\ Colin Barker, Jul 02 2016

Crossrefs

Cf. A000170, A004396, A287864.

Sequence in context: A156351 A057561 A064726 * A257175 A210357 A057359

Adjacent sequences: A274613 A274614 A274615 * A274617 A274618 A274619

Keyword

nonn,easy

Author

Rob Pratt and N. J. A. Sloane, Jul 01 2016

Status

approved